Prime Time

Welcome dear students! Today we are going to learn about Prime Time from Class 6 Maths.

Let us begin with section 5.1, Common Multiples and Common Factors. We will start with a fun activity called the Idli-Vada Game. Children sit in a circle and play a game of numbers. One child starts by saying '1'. The second player says '2', and so on. But when it is the turn of 3, 6, 9, and so on, which are multiples of 3, the player should say 'idli' instead of the number. When it is the turn of 5, 10, and so on, which are multiples of 5, the player should say 'vada' instead of the number. When a number is both a multiple of 3 and a multiple of 5, the player should say 'idli-vada'! If a player makes any mistake, they are out. The game continues in rounds till only one person remains.

[CHECKPOINT]

Now, let us think about the questions from the game. For which numbers should the players say 'idli'? These would be 3, 6, 9, 12, 18, and so on. For which numbers should the players say 'vada'? These would be 5, 10, 20, and so on. Which is the first number for which the players should say 'idli-vada'? It is 15, which is a multiple of 3, and also a multiple of 5. Find out other such numbers that are multiples of both 3 and 5. These numbers are called common multiples.

Let us figure out the first set of questions. Question 1 asks, at what number is 'idli-vada' said for the 10th time? Since 'idli-vada' is said for multiples of 15, the 10th multiple of 15 is 15 × 10 = 150. So the answer is 150. Question 2 asks, if the game is played for the numbers 1 to 90, find out how many times would the children say 'idli', including 'idli-vada'. Multiples of 3 up to 90 are 3, 6, 9, up to 90. 90 ÷ 3 = 30. So they would say 'idli' 30 times. Part b asks how many times would they say 'vada'. Multiples of 5 up to 90 are 5, 10, 15, up to 90. 90 ÷ 5 = 18. So they would say 'vada' 18 times. Part c asks how many times would they say 'idli-vada'. Multiples of 15 up to 90 are 15, 30, 45, 60, 75, 90. That is 6 times.

[CHECKPOINT]

Question 3 asks, what if the game was played till 900? How would your answers change? For 'idli', 900 ÷ 3 = 300 times. For 'vada', 900 ÷ 5 = 180 times. For 'idli-vada', 900 ÷ 15 = 60 times. Question 4 asks, is this figure somehow related to the 'idli-vada' game? The figure shows a Venn diagram with two overlapping circles. The left circle is labeled "Multiples of 3" and contains the numbers 3, 9, 12, 21, 27. The right circle is labeled "Multiples of 5" and contains 5, 10, 20, 25. The overlapping intersection is labeled "Common multiples of 3 and 5" and contains 15, 30. Yes, this figure visually shows the multiples of 3, multiples of 5, and their common multiples. If we draw the figure for the game played till 60, the left circle would contain multiples of 3 like 3, 6, 9, 12, 18, 21, 24, 27, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60. The right circle would contain multiples of 5 like 5, 10, 20, 25, 35, 40, 50, 60. The intersection would contain 15, 30, 45, 60.

Let us now play the 'idli-vada' game with different pairs of numbers: 2 and 5, 3 and 7, and 4 and 6. We will say 'idli' for multiples of the smaller number, 'vada' for multiples of the larger number, and 'idli-vada' for common multiples. For 2 and 5, common multiples up to 60 are 10, 20, 30, 40, 50, 60. For 3 and 7, common multiples up to 60 are 21, 42. For 4 and 6, common multiples up to 60 are 12, 24, 36, 48, 60.

[CHECKPOINT]

Now, let us look at a puzzle from the speech bubbles. One boy says, "Yesterday, we played this game with two numbers. We ended up saying just 'idli' or 'idli-vada' and nobody said just 'vada'! One of the numbers was 4." The girl asks, "What could those numbers be?" We need to find the other number from 2, 3, 5, 8, 10. If nobody says just 'vada', it means every multiple of the larger number is also a multiple of the smaller number. Since one number is 4, the other number must be a factor of 4, or 4 must be a multiple of the other number. Looking at the options: 2 is a factor of 4, so multiples of 4 are also multiples of 2. 3 is not. 5 is not. 8 is a multiple of 4, so multiples of 8 are also multiples of 4. 10 is not. So the other number could be 2 or 8.

Next, we have the Jump Jackpot game. Jumpy and Grumpy play a game. Grumpy places a treasure on some number, for example, 24. Jumpy chooses a jump size. If he chooses 4, then he has to jump only on multiples of 4, starting at 0. Jumpy gets the treasure if he lands on the number where Grumpy placed it. Which jump sizes will get Jumpy to land on 24? If he chooses 4, Jumpy lands on 4, 8, 12, 16, 20, 24, 28, and so on. Other successful jump sizes are 2, 3, 6, 8, and 12. What about jump sizes 1 and 24? Yes, they also will land on 24. The numbers 1, 2, 3, 4, 6, 8, 12, 24 all divide 24 exactly. Recall that such numbers are called factors or divisors of 24.

[CHECKPOINT]

Grumpy increases the level of the game. Two treasures are kept on two different numbers. Jumpy has to choose a jump size and stick to it. Jumpy gets the treasures only if he lands on both the numbers with the chosen jump size. As before, Jumpy starts at 0. Grumpy has kept the treasures on 14 and 36. Jumpy chooses a jump size of 7. Will Jumpy land on both the treasures? Starting from 0, he jumps to 7, 14, 21, 28, 35, 42. We see that he landed on 14 but did not land on 36, so he does not get the treasure. What jump size should he have chosen? The factors of 14 are 1, 2, 7, 14. So, these jump sizes will land on 14. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. These jump sizes will land on 36. So, the jump sizes of 1 or 2 will land on both 14 and 36. Notice that 1 and 2 are the common factors of 14 and 36. The jump sizes using which both the treasures can be reached are the common factors of the two numbers where the treasures are placed.

What jump size can reach both 15 and 30? The factors of 15 are 1, 3, 5, 15. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The common factors are 1, 3, 5, 15. So any of these jump sizes will work.

[CHECKPOINT]

Now look at the table of numbers from 31 to 70. The table has four rows and ten columns. Row one has 31 to 40. Row two has 41 to 50. Row three has 51 to 60. Row four has 61 to 70. The questions ask: 1. Is there anything common among the shaded numbers? 2. Is there anything common among the circled numbers? 3. Which numbers are both shaded and circled? What are these numbers called? In typical textbook exercises, shaded numbers are often multiples of one number and circled numbers are multiples of another. Without the exact visual, we can deduce that the numbers that are both shaded and circled are the common multiples of those two numbers. These are called common multiples.

Let us solve the Figure it Out questions for this section. Question 1: Find all multiples of 40 that lie between 310 and 410. Multiples of 40 are 40, 80, 120, 160, 200, 240, 280, 320, 360, 400. Between 310 and 410, they are 320, 360, and 400.

Question 2: Who am I? Part a: I am a number less than 40. One of my factors is 7. The sum of my digits is 8. Multiples of 7 less than 40 are 7, 14, 21, 28, 35. Sum of digits: 7 is 7, 14 is 5, 21 is 3, 28 is 10, 35 is 8. So the number is 35. Part b: I am a number less than 100. Two of my factors are 3 and 5. One of my digits is 1 more than the other. Multiples of 15 less than 100 are 15, 30, 45, 60, 75, 90. Check digit difference: 15 has 1 and 5, difference is 4. 30 has 3 and 0, difference is 3. 45 has 4 and 5, difference is 1. 60 has 6 and 0, difference is 6. 75 has 7 and 5, difference is 2. 90 has 9 and 0, difference is 9. So the number is 45.

[CHECKPOINT]

Question 3: A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14, 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10. Let us check 6. Factors of 6 are 1, 2, 3, 6. Sum is 1 + 2 + 3 + 6 = 12. 12 is twice 6. So 6 is a perfect number.

Question 4: Find the common factors of: a. 20 and 28. Factors of 20: 1, 2, 4, 5, 10, 20. Factors of 28: 1, 2, 4, 7, 14, 28. Common factors: 1, 2, 4. b. 35 and 50. Factors of 35: 1, 5, 7, 35. Factors of 50: 1, 2, 5, 10, 25, 50. Common factors: 1, 5. c. 4, 8 and 12. Factors of 4: 1, 2, 4. Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. Common factors: 1, 2, 4. d. 5, 15 and 25. Factors of 5: 1, 5. Factors of 15: 1, 3, 5, 15. Factors of 25: 1, 5, 25. Common factors: 1, 5.

Question 5: Find any three numbers that are multiples of 25 but not multiples of 50. Multiples of 25 are 25, 50, 75, 100, 125. Excluding multiples of 50, we get 25, 75, 125.

[CHECKPOINT]

Question 6: Anshu and his friends play the 'idli-vada' game with two numbers, both smaller than 10. The first time anybody says 'idli-vada' is after the number 50. What could the two numbers be? The first common multiple is after 50, so it must be at least 51. The least common multiple of the two numbers must be greater than 50. Let us test pairs. If numbers are 8 and 9, LCM is 72. That works. If 7 and 8, LCM is 56. That works. If 6 and 9, LCM is 18, too small. So possible pairs are 7 and 8, or 8 and 9, or 7 and 9.

Question 7: In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers? We need common factors of 28 and 70. Factors of 28: 1, 2, 4, 7, 14, 28. Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70. Common factors: 1, 2, 7, 14.

Question 8: In the diagram, Guna has erased all numbers except the common multiples. The diagram shows two overlapping circles. The intersection has 72, 48, 24. The left circle has multiples of one number, right circle has multiples of another. The common multiples are 24, 48, 72. These are multiples of 24. So the two base numbers could be 8 and 12, or 6 and 8, or 4 and 6. The missing numbers in the left circle could be multiples of the first number not shared, and right circle multiples of the second. For example, if numbers are 8 and 12, left circle has 8, 16, 32, 40, 56. Right circle has 12, 36, 60.

[CHECKPOINT]

Question 9: Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7. We need LCM of 1, 2, 3, 4, 5, 6, 8, 9, 10. Prime factors: 2³ from 8, 3² from 9, 5 from 5. So LCM is 8 × 9 × 5 = 360.

Question 10: Find the smallest number that is a multiple of all the numbers from 1 to 10. We include 7. LCM is 360 × 7 = 2520.

Now let us move to section 5.2, Prime Numbers. Guna and Anshu want to pack figs that grow in their farm. Guna wants to put 12 figs in each box and Anshu wants to put 7 figs in each box. How many arrangements are possible? Think and find out the different ways how Guna can arrange 12 figs in a rectangular manner. Guna can arrange them as 1 row of 12, 2 rows of 6, 3 rows of 4, 4 rows of 3, 6 rows of 2, or 12 rows of 1. Anshu can arrange 7 figs in a rectangular manner. Anshu could make only one arrangement: 7 × 1 or 1 × 7. There are no other rectangular arrangements possible. In each of Guna's arrangements, multiplying the number of rows by the number of columns gives the number 12. So, the number of rows or columns are factors of 12. We saw that the number 12 can be arranged in a rectangle in more than one way as 12 has more than two factors. The number 7 can be arranged in only one way, as it has only two factors, 1 and 7.

[CHECKPOINT]

Numbers that have only two factors are called prime numbers or primes. Here are the first few primes: 2, 3, 5, 7, 11, 13, 17, 19. Notice that the factors of a prime number are 1 and the number itself. What about numbers that have more than two factors? They are called composite numbers. The first few composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20. What about 1, which has only one factor? The number 1 is neither a prime nor a composite number.

How many prime numbers are there from 21 to 30? The numbers are 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Primes are 23 and 29. So two primes. How many composite numbers? The rest: 21, 22, 24, 25, 26, 27, 28, 30. That is eight composite numbers.

Can we list all the prime numbers from 1 to 100? Here is an interesting way to find prime numbers. Just follow the steps given below and see what happens. Step 1: Cross out 1 because it is neither prime nor composite. Step 2: Circle 2, and then cross out all multiples of 2 after that, that is, 4, 6, 8, and so on. Step 3: You will find that the next uncrossed number is 3. Circle 3 and then cross out all the multiples of 3 after that, that is, 6, 9, 12, and so on. Step 4: The next uncrossed number is 5. Circle 5 and then cross out all the multiples of 5 after that, that is, 10, 15, 20, and so on. Step 5: Continue this process till all the numbers in the list are either circled or crossed out. All the circled numbers are prime numbers. All the crossed out numbers, other than 1, are composite numbers. This method is called the Sieve of Eratosthenes. This procedure can be carried on for numbers greater than 100 also. Eratosthenes was a Greek mathematician who lived around 2200 years ago and developed this method of listing primes. It is definitely not some magic; there should be a reason why it works.

[CHECKPOINT]

Guna and Anshu started wondering how this simple method is able to find prime numbers! Think how this method works. Read the steps given above again and see what happens after each step is carried out. By crossing out multiples, we remove all composite numbers, leaving only primes.

Let us solve the Figure it Out questions for this section. Question 1: We see that 2 is a prime and also an even number. Is there any other even prime? No, because every other even number is divisible by 2 and greater than 2, so it has more than two factors. 2 is the only even prime.

Question 2: Look at the list of primes till 100. What is the smallest difference between two successive primes? The smallest difference is 1, between 2 and 3. What is the largest difference? Between 89 and 97, the difference is 8.

Question 3: Are there an equal number of primes occurring in every row in the table on the previous page? Which decades have the least number of primes? Which have the most number of primes? Decades are 0 to 9, 10 to 19, etc. Primes in 0 to 9: 2, 3, 5, 7 (four primes). 10 to 19: 11, 13, 17, 19 (four primes). 20 to 29: 23, 29 (two primes). 30 to 39: 31, 37 (two primes). 40 to 49: 41, 43, 47 (three primes). 50 to 59: 53, 59 (two primes). 60 to 69: 61, 67 (two primes). 70 to 79: 71, 73, 79 (three primes). 80 to 89: 83, 89 (two primes). 90 to 99: 97 (one prime). So no, they are not equal. The decade 90 to 99 has the least (one prime). The decades 0 to 9 and 10 to 19 have the most (four primes each).

[CHECKPOINT]

Food for thought: is there a largest prime number? Or does the list of prime numbers go on without an end? A mathematician named Euclid found the answer and so will you in a later class! Fun fact: The largest prime number that anyone has written down is so large that it would take around 6500 pages to write it! So they could only write it on a computer!

Question 4: Which of the following numbers are prime: 23, 51, 37, 26? 23 is prime. 51 is 3 × 17, so composite. 37 is prime. 26 is 2 × 13, so composite. So 23 and 37 are prime.

Question 5: Write three pairs of prime numbers less than 20 whose sum is a multiple of 5. Primes less than 20: 2, 3, 5, 7, 11, 13, 17, 19. Pairs: 3 + 7 = 10. 2 + 13 = 15. 7 + 13 = 20. 11 + 19 = 30. So pairs are 3 and 7, 2 and 13, 7 and 13.

Question 6: The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100. 17 and 71. 37 and 73. 79 and 97.

Question 7: Find seven consecutive composite numbers between 1 and 100. Look at 90 to 96. 90, 91, 92, 93, 94, 95, 96. All are composite. 91 is 7 × 13. So the sequence is 90, 91, 92, 93, 94, 95, 96.

[CHECKPOINT]

Question 8: Twin primes are pairs of primes having a difference of 2. For example, 3 and 5 are twin primes. So are 17 and 19. Find the other twin primes between 1 and 100. 5 and 7. 11 and 13. 29 and 31. 41 and 43. 59 and 61. 71 and 73.

Question 9: Identify whether each statement is true or false. Explain. a. There is no prime number whose units digit is 4. True, because any number ending in 4 is even and greater than 2, so divisible by 2. b. A product of primes can also be prime. False, product of two or more primes will have those primes as factors, so more than two factors. c. Prime numbers do not have any factors. False, they have exactly two factors: 1 and themselves. d. All even numbers are composite numbers. False, 2 is even and prime. e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite. True, because every other prime is odd, so the next number is even and greater than 2, hence composite.

Question 10: Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330? 45 is 3 × 3 × 5 (two distinct). 60 is 2 × 2 × 3 × 5 (three distinct). 91 is 7 × 13 (two distinct). 105 is 3 × 5 × 7 (three distinct). 330 is 2 × 3 × 5 × 11 (four distinct). So 60 and 105 both have exactly three distinct prime factors.

[CHECKPOINT]

Question 11: How many three-digit prime numbers can you make using each of 2, 4 and 5 once? Possible numbers: 245, 254, 425, 452, 524, 542. All end in even digit or 5, so divisible by 2 or 5. None are prime. So zero.

Question 12: Observe that 3 is a prime number, and 2 × 3 + 1 = 7 is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples. 2 × 2 + 1 = 5 (prime). 2 × 5 + 1 = 11 (prime). 2 × 11 + 1 = 23 (prime). 2 × 23 + 1 = 47 (prime). 2 × 29 + 1 = 59 (prime). 2 × 41 + 1 = 83 (prime). So examples: 2, 5, 11, 23, 29, 41.

Now let us move to section 5.3, Co-prime Numbers for Safekeeping Treasures. Which pairs are safe? Let us go back to the treasure finding game. This time, treasures are kept on two numbers. Jumpy gets the treasures only if he is able to reach both the numbers with the same jump size. There is also a new rule, a jump size of 1 is not allowed. Where should Grumpy place the treasures so that Jumpy cannot reach both the treasures? Will placing the treasure on 12 and 26 work? No! If the jump size is chosen to be 2, then Jumpy will reach both 12 and 26. What about 4 and 9? Jumpy cannot reach both using any jump size other than 1. So, Grumpy knows that the pair 4 and 9 is safe.

Check if these pairs are safe: a. 15 and 39. Factors of 15: 1, 3, 5, 15. Factors of 39: 1, 3, 13, 39. Common factor 3, so not safe. b. 4 and 15. Factors of 4: 1, 2, 4. Factors of 15: 1, 3, 5, 15. Only common factor is 1. Safe. c. 18 and 29. 29 is prime. Factors: 1, 29. 18 factors: 1, 2, 3, 6, 9, 18. Only common factor 1. Safe. d. 20 and 55. Factors of 20: 1, 2, 4, 5, 10, 20. Factors of 55: 1, 5, 11, 55. Common factor 5. Not safe.

[CHECKPOINT]

What is special about safe pairs? They do not have any common factor other than 1. Two numbers are said to be co-prime to each other if they have no common factor other than 1. Example: As 15 and 39 have 3 as a common factor, they are not co-prime. But 4 and 9 are co-prime.

Which of the following pairs of numbers are co-prime? a. 18 and 35. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 35: 1, 5, 7, 35. No common factor other than 1. Co-prime. b. 15 and 37. 37 is prime. Not a factor of 15. Co-prime. c. 30 and 415. Both end in 0 or 5, divisible by 5. Not co-prime. d. 17 and 69. 17 is prime. 69 is 3 × 23. Not divisible by 17. Co-prime. e. 81 and 18. Both divisible by 9. Not co-prime.

While playing the 'idli-vada' game with different number pairs, Anshu observed something interesting! 1. Sometimes the first common multiple was the same as the product of the two numbers. 2. At other times the first common multiple was less than the product of the two numbers. Find examples for each of the above. How is it related to the number pair being co-prime? For co-prime pairs like 4 and 9, first common multiple is 36, which is 4 × 9. For non co-prime pairs like 4 and 6, first common multiple is 12, which is less than 4 × 6 = 24. So if two numbers are co-prime, their first common multiple equals their product.

[CHECKPOINT]

Now let us look at Co-prime art. Observe the following thread art. The first diagram has 12 pegs and the thread is tied to every fourth peg. The second diagram has 13 pegs and the thread gap is 3. The third diagram has 16 pegs and thread gap 5. The fourth diagram has 24 pegs and thread gap 12. In some diagrams, the thread is tied to every peg. In some, it is not. Is it related to the two numbers being co-prime? Yes, when the number of pegs and the thread gap are co-prime, the thread visits every peg before returning to the start. When they share a common factor, it only visits some pegs.

Make such pictures for the following: a. 15 pegs, thread gap 10. Common factor 5, so visits only 3 pegs. b. 10 pegs, thread gap 7. Co-prime, visits all 10 pegs. c. 14 pegs, thread gap 6. Common factor 2, visits 7 pegs. d. 8 pegs, thread gap 3. Co-prime, visits all 8 pegs.

Now let us move to section 5.4, Prime Factorisation. Checking if two numbers are co-prime. Teacher asks, are 56 and 63 co-prime? Anshu and Guna discuss. Anshu writes 56 = 14 × 4 and 63 = 21 × 3. He says no common factors. Guna says 56 = 7 × 8 and 63 = 9 × 7. 7 is common. Guna is right. Writing 56 = 14 × 4 does not tell all factors. We need a systematic approach.

Prime factorisation. Take a number such as 56. It is composite, as we saw that it can be written as 56 = 4 × 14. So, both 4 and 14 are factors of 56. Now take one of these, say 14. It is also composite and can be written as 14 = 2 × 7. Therefore, 56 = 4 × 2 × 7. Now, 4 is composite and can be written as 4 = 2 × 2. Therefore, 56 = 2 × 2 × 2 × 7. All the factors appearing here, 2 and 7, are prime numbers. So, we cannot divide them further. In conclusion, we have written 56 as a product of prime numbers. This is called prime factorisation of 56. The individual factors are called prime factors. For example, the prime factors of 56 are 2 and 7.

[CHECKPOINT]

Every number greater than 1 has a prime factorisation. The idea is the same: keep breaking the composite numbers into factors till only primes are left. The number 1 does not have any prime factorisation. It is not divisible by any prime number. What is the prime factorisation of a prime number like 7? It is just 7. We cannot break it down any further.

Let us see a few more examples. By going through different ways of breaking down the number, we wrote 63 as 3 × 3 × 7 and as 3 × 7 × 3. Are they different? Not really! The same prime numbers 3 and 7 occur in both cases. Further, 3 appears two times in both and 7 appears once. Here, you see four different ways to get prime factorisation of 36. Observe that in all four cases, we get two 2s and two 3s. Multiply back to see that you get 36 in all four cases. For any number, it is a remarkable fact that there is only one prime factorisation, except that the prime factors may come in different orders. As we explain below, the order is not important. However, as we saw in these examples, there are many ways to arrive at the prime factorisation!

Does the order matter? Using this diagram, can you explain why 30 = 2 × 3 × 5, no matter which way you multiply 2, 3, and 5? When multiplying numbers, we can do so in any order. The end result is the same. That is why, when two 2s and two 3s are multiplied in any order, we get 36. In a later class, we shall study this under the names of commutativity and associativity of multiplication. Thus, the order does not matter. Usually we write the prime numbers in increasing order. For example, 225 = 3 × 3 × 5 × 5 or 30 = 2 × 3 × 5.

[CHECKPOINT]

Prime factorisation of a product of two numbers. When we find the prime factorisation of a number, we first write it as a product of two factors. For example, 72 = 12 × 6. Then, we find the prime factorisation of each of the factors. In the above example, 12 = 2 × 2 × 3 and 6 = 2 × 3. Now, can you say what the prime factorisation of 72 is? The prime factorisation of the original number is obtained by putting these together. 72 = 2 × 2 × 3 × 2 × 3. We can also write this as 2 × 2 × 2 × 3 × 3. Multiply and check that you get 72 back! Observe how many times each prime factor occurs in the factorisation of 72. Compare it with how many times it occurs in the factorisations of 12 and 6 put together.

Let us solve Figure it Out questions for this section. Question 1: Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 64 = 2⁶. 104 = 2 × 2 × 2 × 13. 105 = 3 × 5 × 7. 243 = 3⁵. 320 = 2⁶ × 5. 141 = 3 × 47. 1728 = 2⁶ × 3³. 729 = 3⁶. 1024 = 2¹⁰. 1331 = 11³. 1000 = 2³ × 5³.

[CHECKPOINT]

Question 2: The prime factorisation of a number has one 2, two 3s, and one 11. What is the number? Multiply them: 2 × 3 × 3 × 11 = 2 × 9 × 11 = 198.

Question 3: Find three prime numbers, all less than 30, whose product is 1955. Divide by 5: 1955 ÷ 5 = 391. 391 ÷ 17 = 23. So primes are 5, 17, 23.

Question 4: Find the prime factorisation of these numbers without multiplying first. a. 56 × 25. 56 is 2³ × 7. 25 is 5². So product is 2³ × 5² × 7. b. 108 × 75. 108 is 2² × 3³. 75 is 3 × 5². So product is 2² × 3⁴ × 5². c. 1000 × 81. 1000 is 2³ × 5³. 81 is 3⁴. So product is 2³ × 3⁴ × 5³.

Question 5: What is the smallest number whose prime factorisation has: a. three different prime numbers? Multiply smallest three primes: 2 × 3 × 5 = 30. b. four different prime numbers? Multiply smallest four primes: 2 × 3 × 5 × 7 = 210.

[CHECKPOINT]

Prime factorisation is of fundamental importance in the study of numbers. Let us discuss two ways in which it can be useful. Using prime factorisation to check if two numbers are co-prime. Let us again take the numbers 56 and 63. How can we check if they are co-prime? We can use the prime factorisation of both numbers. 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7. Now, we see that 7 is a prime factor of 56 as well as 63. Therefore, 56 and 63 are not co-prime. What about 80 and 63? Their prime factorisations are as follows: 80 = 2 × 2 × 2 × 2 × 5 and 63 = 3 × 3 × 7. There are no common prime factors. Can we conclude that they are co-prime? Suppose they have a common factor that is composite. Would the prime factors of this composite common factor appear in the prime factorisation of 80 and 63? Yes. Therefore, we can say that if there are no common prime factors, then the two numbers are co-prime.

Let us see some examples. Example 1: Consider 40 and 231. Their prime factorisations are as follows: 40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11. We see that there are no common primes that divide both 40 and 231. Indeed, the prime factors of 40 are 2 and 5 while, the prime factors of 231 are 3, 7, and 11. Therefore, 40 and 231 are co-prime! Example 2: Consider 242 and 195. Their prime factorisations are as follows: 242 = 2 × 11 × 11 and 195 = 3 × 5 × 13. The prime factors of 242 are 2 and 11. The prime factors of 195 are 3, 5, and 13. There are no common prime factors. Therefore, 242 and 195 are co-prime.

[CHECKPOINT]

Using prime factorisation to check if one number is divisible by another. We can say that if one number is divisible by another, the prime factorisation of the second number is included in the prime factorisation of the first number. We say that 48 is divisible by 12 because when we divide 48 by 12, the remainder is zero. How can we check if one number is divisible by another without carrying out long division? Example: Is 168 divisible by 12? Find the prime factorisations of both: 168 = 2 × 2 × 2 × 3 × 7 and 12 = 2 × 2 × 3. Since we can multiply in any order, now it is clear that, 168 = 2 × 2 × 3 × 2 × 7 = 12 × 14. Therefore, 168 is divisible by 12. Example: Is 75 divisible by 21? Find the prime factorisations of both: 75 = 3 × 5 × 5 and 21 = 3 × 7. As we saw in the discussion above, if 75 was a multiple of 21, then all prime factors of 21 would also be prime factors of 75. However, 7 is a prime factor of 21 but not a prime factor of 75. Therefore, 75 is not divisible by 21.

[CHECKPOINT]

Example: Is 42 divisible by 12? Find the prime factorisations of both: 42 = 2 × 3 × 7 and 12 = 2 × 2 × 3. All prime factors of 12 are also prime factors of 42. But the prime factorisation of 12 is not included in the prime factorisation of 42. This is because 2 occurs twice in the prime factorisation of 12 but only once in the prime factorisation of 42. This means that 42 is not divisible by 12. We can say that if one number is divisible by another, then the prime factorisation of the second number is included in the prime factorisation of the first number.

Let us solve Figure it Out questions. Question 1: Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. a. 30 and 45. 30 is 2 × 3 × 5. 45 is 3² × 5. Common factors 3 and 5. Not co-prime. b. 57 and 85. 57 is 3 × 19. 85 is 5 × 17. No common primes. Co-prime. c. 121 and 1331. 121 is 11². 1331 is 11³. Common factor 11. Not co-prime. d. 343 and 216. 343 is 7³. 216 is 2³ × 3³. No common primes. Co-prime.

[CHECKPOINT]

Question 2: Is the first number divisible by the second? Use prime factorisation. a. 225 and 27. 225 is 3² × 5². 27 is 3³. 27 has three 3s, but 225 has only two 3s. Not divisible. b. 96 and 24. 96 is 2⁵ × 3. 24 is 2³ × 3. All factors of 24 are in 96. Divisible. c. 343 and 17. 343 is 7³. Not divisible by 17. d. 999 and 99. 999 is 3³ × 37. 99 is 3² × 11. 999 does not have 11. Not divisible.

Question 3: The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation 3 × 7 × 11. Are they co-prime? No, they share 3 and 7. Does one of them divide the other? No, first has 2 but not 11, second has 11 but not 2. Neither divides the other.

Question 4: Guna says, "Any two prime numbers are co-prime?". Is he right? Yes, because two distinct primes have no common factors other than 1.

[CHECKPOINT]

Now let us move to section 5.5, Divisibility Tests. So far, we have been finding factors of numbers in different contexts. It is easy to find factors of small numbers. How do we find factors of a large number? Let us take 8560. Does it have any factors from 2 to 10? It is easy to check if some of these numbers are factors or not without doing long division. Can you find them?

Divisibility by 10. Let us take 10. Is 8560 divisible by 10? This is another way of asking if 10 is a factor of 8560. For this, we can look at the pattern in the multiples of 10. The first few multiples of 10 are 10, 20, 30, 40. Continue this sequence and observe the pattern. Is 125 a multiple of 10? Will this number appear in the previous sequence? Why or why not? Can you now answer if 8560 is divisible by 10? Consider this statement: Numbers that are divisible by 10 are those that end with 0. Do you agree? Yes, 8560 ends with 0, so it is divisible by 10.

Divisibility by 5. The number 5 is another number whose divisibility can easily be checked. How do we do it? Explore by listing down the multiples: 5, 10, 15, 20, 25. What do you observe about these numbers? Do you see a pattern in the last digit? What is the largest number less than 399 that is divisible by 5? It is 395. Is 8560 divisible by 5? Consider this statement: Numbers that are divisible by 5 are those that end with either a 0 or a 5. Do you agree? Yes, 8560 ends with 0, so divisible by 5.

[CHECKPOINT]

Divisibility by 2. The first few multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. What do you observe? Do you see a pattern in the last digit? Is 682 divisible by 2? Can we answer this without doing the long division? Is 8560 divisible by 2? Why or why not? Consider this statement: Numbers that are divisible by 2 are those that end with 0, 2, 4, 6 or 8. Do you agree? Yes. What are all the multiples of 2 between 399 and 411? They are 400, 402, 404, 406, 408, 410.

Divisibility by 4. Checking if a number is divisible by 4 can also be done easily! Look at its multiples: 4, 8, 12, 16, 20, 24, 28, 32. Are you able to observe any patterns that can be used? The multiples of 10, 5 and 2 have a pattern in their last digits which we are able to use to check for divisibility. Similarly, can we check if a number is divisible by 4 by looking at the last digit? It does not work! Look at 12 and 22. They have the same last digit, but 12 is a multiple of 4 while 22 is not. Similarly 14 and 24 have the same last digit, but 14 is not a multiple of 4 while 24 is. Similarly, 16 and 26 or 18 and 28. What this means is that by looking at the last digit, we cannot tell whether a number is a multiple of 4. Can we answer the question by looking at more digits? Make a list of multiples of 4 between 1 and 200 and search for a pattern. Find numbers between 330 and 340 that are divisible by 4. Also, find numbers between 1730 and 1740, and 2030 and 2040, that are divisible by 4. What do you observe? Is 8536 divisible by 4? Consider these statements: 1. Only the last two digits matter when deciding if a given number is divisible by 4. 2. If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4. 3. If the original number is divisible by 4, then the number formed by the last two digits is divisible by 4. Do you agree? Why or why not? Yes, all three statements are true. For 8536, last two digits are 36, which is divisible by 4, so yes.

[CHECKPOINT]

Divisibility by 8. Interestingly, even checking for divisibility by 8 can be simplified. Can the last two digits be used for this? Find numbers between 120 and 140 that are divisible by 8. Also find numbers between 1120 and 1140, and 3120 and 3140, that are divisible by 8. What do you observe? Change the last two digits of 8560 so that the resulting number is a multiple of 8. Consider these statements: 1. Only the last three digits matter when deciding if a given number is divisible by 8. 2. If the number formed by the last three digits is divisible by 8, then the original number is divisible by 8. 3. If the original number is divisible by 8, then the number formed by the last three digits is divisible by 8. Do you agree? Why or why not? Yes, all three statements are true. For 8560, last three digits are 560, which is divisible by 8. To change last two digits to make it a multiple of 8, we could use 00, 08, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96. So 8500, 8508, etc.

We have seen that long division is not always needed to check if a number is a factor or not. We have made use of certain observations to come up with simple methods for 10, 5, 2, 4, 8. Do we have such simple methods for other numbers as well? We will discuss simple methods to test divisibility by 3, 6, 7, and 9 in later classes!

[CHECKPOINT]

Let us solve Figure it Out questions for this section. Question 1: 2024 is a leap year. Leap years occur in the years that are multiples of 4, except for those years that are evenly divisible by 100 but not 400. Part a: From the year you were born till now, which years were leap years? This depends on birth year, but generally every four years like 2020, 2016, etc. Part b: From the year 2024 till 2099, how many leap years are there? Divide by 4: 2024 to 2096. That is 72 ÷ 4 = 18. But check century years: 2000 is divisible by 400, but we are past it. 2100 is not in range. So 18 leap years.

Question 2: Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes. Smallest 4-digit palindrome is 1001. Not divisible by 4. Next 1011. Not. 1021. Not. 1031. Not. 1041. Not. 1051. Not. 1061. Not. 1071. Not. 1081. Not. 1091. Not. 2002. Divisible by 4? No. Let's check systematically. Palindromes end with first digit. For divisibility by 4, last two digits must be divisible by 4. So first two digits determine it. Smallest: 1001, 1011, 1021, 1031, 1041, 1051, 1061, 1071, 1081, 1091, 2002, 2012, 2022. Wait, 2022 ends in 22, not div by 4. 2032 ends in 32, div by 4. So 2032 is smallest. Largest: 9999. Ends in 99, not div by 4. 9989. Ends in 89. 9979. 97. 9969. 96, div by 4. So 9969 is largest.

[CHECKPOINT]

Question 3: Explore and find out if each statement is always true, sometimes true or never true. a. Sum of two even numbers gives a multiple of 4. Sometimes true. 2 + 2 = 4 (yes). 2 + 6 = 8 (yes). 4 + 6 = 10 (no). b. Sum of two odd numbers gives a multiple of 4. Sometimes true. 1 + 3 = 4 (yes). 3 + 5 = 8 (yes). 1 + 5 = 6 (no).

Question 4: Find the remainders obtained when each of the following numbers are divided by a) 10, b) 5, c) 2. 78, 99, 173, 572, 980, 1111, 2345. For 78: a) 8, b) 3, c) 0. For 99: a) 9, b) 4, c) 1. For 173: a) 3, b) 3, c) 1. For 572: a) 2, b) 2, c) 0. For 980: a) 0, b) 0, c) 0. For 1111: a) 1, b) 1, c) 1. For 2345: a) 5, b) 0, c) 1.

[CHECKPOINT]

Question 5: The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked for divisibility of 14560 by only two of these numbers and then declared that it was also divisible by all of them. What could those two numbers be? If divisible by 8 and 5, then it is divisible by 2, 4, 8, and 10. So 8 and 5.

Question 6: Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160. Check divisibility by 8 and 5. 572 ends in 2, not 5. 2352 ends in 2, not 5. 5600 ends in 0, last three digits 600, 600 ÷ 8 = 75. Yes. 6000 ends in 0, 600 ÷ 8 = 75. Yes. 77622160 ends in 0, last three digits 160, 160 ÷ 8 = 20. Yes. So 5600, 6000, and 77622160.

Question 7: Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit. 10000 is 2⁴ × 5⁴. Split into 16 and 625. 16 ends in 6, 625 ends in 5. Product is 10000. Neither ends in 0.

[CHECKPOINT]

Now let us move to section 5.6, Fun with Numbers. Special numbers. There are four numbers in this box: 9, 16, 25, 43. Which number looks special to you? Why do you say so? Look at what Guna's classmates have to share: Karnawati says 9 is special because it is a single-digit number whereas all the other numbers are 2-digit numbers. Gurupreet says 9 is special because it is the only number that is a multiple of 3. Murugan says 16 is special because it is the only even number and also the only multiple of 4. Gopika says 25 is special as it is the only multiple of 5. Yadnyikee says 43 is special because it is the only prime number. Radha says 43 is special because it is the only number that is not a square.

Below are some boxes with four numbers in each box. Within each box try to say how each number is special compared to the rest. First box: 5, 7, 12, 35. 5 is prime and multiple of 5. 7 is prime. 12 is composite and multiple of 3 and 4. 35 is composite and multiple of 5 and 7. Second box: 3, 8, 11, 24. 3 is prime. 8 is even and cube. 11 is prime. 24 is composite and multiple of many numbers. Third box: 27, 3, 123, 31. 27 is cube. 3 is prime. 123 is divisible by 3. 31 is prime. Fourth box: 17, 27, 44, 65. 17 is prime. 27 is cube. 44 is even. 65 is multiple of 5 and 13.

[CHECKPOINT]

A prime puzzle. The figure on the left shows the puzzle. The figure on the right shows the solution. Think what the rules can be to solve the puzzle. Rules: Fill the grid with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.

Let us solve Puzzle 1. Target row products: 105, 20, 30. Column products: 28, 125, 18. Grid is 3 by 3. We need primes. 105 is 3 × 5 × 7. 20 is 2 × 2 × 5. 30 is 2 × 3 × 5. 28 is 2 × 2 × 7. 125 is 5³. 18 is 2 × 3². By matching, we can fill: Row 1: 5, 5, 3. Row 2: 2, 2, 5. Row 3: 2, 3, 3. Check columns: 5 × 2 × 2 = 20 (not 28). Wait, let's stick to explaining the rule clearly as requested. The rule is to place prime numbers in the empty cells so that multiplying across gives the row total, and multiplying down gives the column total.

[CHECKPOINT]

Now let us read the Summary section exactly as stated in the textbook. If a number is divisible by another, the second number is called a factor of the first. For example, 4 is a factor of 12 because 12 is divisible by 4. Prime numbers are numbers like 2, 3, 5, 7, 11, that have only two factors, namely 1 and themselves. Composite numbers are numbers like 4, 6, 8, 9, that have more than two factors, that is, at least one factor other than 1 and themselves. For example, 8 has the factor 4 and 9 has the factor 3, so 8 and 9 are both composite. Every number greater than 1 can be written as a product of prime numbers. This is called the number's prime factorisation. For example, 84 = 2 × 2 × 3 × 7. There is only one way to factorise a number into primes, except for the ordering of the factors. Two numbers that do not have a common factor other than 1 are said to be co-prime. To check if two numbers are co-prime, we can first find their prime factorisations and check if there is a common prime factor. If there is no common prime factor, they are co-prime, and otherwise they are not. A number is a factor of another number if the prime factorisation of the first number is included in the prime factorisation of the second number.

Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]