A multiple contains its number

The idea

The prime factorisation of a multiple of a number contains the number's prime factors along with some more, so every common multiple contains the prime factorisations of all the numbers.

36 and 648

36 = 2 × 2 × 3 × 3. A multiple of 36 is 648 = 36 × 18. Break 18 into primes too: 18 = 2 × 3 × 3.

Look at the primes of 648. Do you think the primes of 36 are inside them?

• Yes, all of them
• Only some of them
• No, they change completely

Look inside the multiples

Choose a multiplier. The ringed beads of the multiple are the beads of 36. Then judge which numbers are common multiples of two numbers.

The beads are always inside

A multiple of 36 is 36 × something. So its prime factorisation has all the primes of 36, and then the primes of the something.

A common multiple of 14 and 35 must contain 14 = 2 × 7 and 35 = 5 × 7. For example, 70 = 2 × 5 × 7 contains both: the single 7 serves both numbers.

The prime factorisation of a multiple of a number contains the number’s prime factors, along with some more. So every common multiple contains the prime factorisations of all the numbers.

• Number | Prime factorisation
• 36 | 2 × 2 × 3 × 3
• 18 | 2 × 3 × 3
• 648 | 2 × 2 × 2 × 3 × 3 × 3 × 3

Notes

The prime factorisation of a multiple of a number contains the number’s prime factors, along with some more. So every common multiple contains the prime factorisations of all the numbers.

Check yourself

12 = 2 × 2 × 3. Which of these is the prime factorisation of a multiple of 12?

45 = 3 × 3 × 5. A multiple of 45 has the prime factorisation 3 × 3 × 5 × 7 × 7. What was 45 multiplied by?

Answer: 49

Taking out 3 × 3 × 5 leaves 7 × 7 = 49. So the multiple is 45 × 49 = 2205.

70 = 2 × 5 × 7 has only one 7. But 14 = 2 × 7 and 35 = 5 × 7 both have a 7. Is 70 a common multiple of 14 and 35?

6 = 2 × 3 and 15 = 3 × 5. Which of these is a common multiple?

• 2 × 3 × 5. It has only one 2. A multiple of 12 needs both 2s and the 3.
• 2 × 2 × 3 × 5 — correct. Yes! It has all the primes of 12 (2 × 2 × 3), and a 5 more. It is 60 = 12 × 5.
• 2 × 2 × 5 × 7. The 3 is missing. A multiple of 12 must contain 2 × 2 × 3.
• No: 70 would need two 7s, one for each number.. The two numbers are checked one at a time. 14 finds 2 × 7 inside 70, and 35 finds 5 × 7 inside 70.
• No: 70 is not a multiple of 35.. 70 = 35 × 2, so it is a multiple of 35.
• Yes: one 7 is enough, because 14 and 35 each only need to find a 7 in 70. — correct. Yes! Each number is checked separately, so they can share the same bead.
• 2 × 3 × 3. It has no 5, so it does not contain 15 = 3 × 5.
• 2 × 3 × 5 — correct. Yes! It contains 2 × 3 (for 6) and 3 × 5 (for 15). It is 30.
• 3 × 5 × 7. It has no 2, so it does not contain 6 = 2 × 3.