The division method

The idea

In the division method, a composite number is repeatedly divided by a prime factor, writing each quotient below, until a prime is reached; the primes collected give the prime factorisation.

Breaking 105

Every number can be written as a product of primes. For 105, one prime that divides it is 3: 105 ÷ 3 = 35. Then 35 can be divided by another prime.

You keep dividing the quotients by primes. When do you think you stop?

• When the quotient is a prime
• When the quotient is odd
• When the quotient is less than 10

Climb down the ladder

Tap a prime to divide. The quotient goes below. A prime that does not fit is refused. You can undo a step.

Primes on the left, a prime at the bottom

Each time the number is divided by a prime, the quotient goes below it. We stop when the quotient is a prime.

The primes on the left (3, 5) and the prime at the bottom (7) are the prime factorisation: 105 = 3 × 5 × 7. Different first primes give the same primes, only the order changes.

Division method: divide a composite number by a prime factor, write the quotient below, and repeat until you reach a prime. The primes collected give the prime factorisation.

• Divide by | Quotient
• 3 | 105 ÷ 3 = 35
• 5 | 35 ÷ 5 = 7, a prime, so stop

Notes

Division method: divide a composite number by a prime factor, writing each quotient below, until you reach a prime. The primes collected give the prime factorisation.

Check yourself

Use the division method on 150. How many times does the prime 5 appear in its prime factorisation?

Answer: 2

150 ÷ 2 = 75, 75 ÷ 3 = 25, 25 ÷ 5 = 5. The primes are 2, 3, 5 and 5, so 5 appears 2 times.

Aarav writes a ladder for 84. The first step is: divide 84 by 4 to get 21. Is this a correct step of the division method?

A ladder for 90 starts: divide by 2 to get 45. Can we stop at 45?

A ladder for a number has the divisors 2, 2, 3 down the left and the prime 7 at the bottom. What is the number?

Answer: 84

The number is the product of all the primes: 2 × 2 × 3 × 7 = 84.

• No: the divisor must be a prime, and 4 is not. Divide by 2 twice instead. — correct. Yes! 4 = 2 × 2, so it takes two steps with the prime 2.
• Yes: 4 divides 84 exactly.. The division is exact, but the method divides only by primes. 4 is a composite number.
• No: 84 cannot be divided by 4.. 84 ÷ 4 = 21 exactly, so it can be divided. The problem is that 4 is not a prime.
• Yes: 45 is odd, so it cannot be divided any more.. Odd numbers can still be composite. 45 is divisible by 3 and by 5.
• Yes: 2 is the only even prime, so we are done.. We stopped dividing by 2, but the bottom number 45 is still composite.
• No: 45 is not a prime (45 = 3 × 15). Keep dividing. — correct. Yes! We stop only when the number at the bottom is a prime.