Factors from the primes

The idea

Every factor of a number is a subpart (a product of some of the primes) of its prime factorisation, so systematically listing all subparts, together with 1, gives all its factors.

Is 28 a factor of 840?

840 = 2 × 2 × 2 × 3 × 5 × 7. Is 28 a factor of 840? We can reorder the primes: 840 = (2 × 2 × 7) × (2 × 3 × 5).

2 × 2 × 7 = 28 is made from some of the primes of 840. Is 28 a factor of 840?

• Yes, and it is multiplied by 30
• No, 28 does not divide 840
• Only if 840 is divisible by 7

Pick beads, find factors

Each bead is a prime. Tap beads to pick a subpart. The beads left over make its partner. Find all the factors of the number.

A factor is a subpart

840 = (2 × 2 × 7) × (2 × 3 × 5) = 28 × 30. So 28 is a factor and 840 is 28 times 30. The beads you pick make one factor, and the beads left over make the other.

To list all the factors of 225 = 3 × 3 × 5 × 5, form the subparts one size at a time:

Add 1 (no primes). The factors of 225 are 1, 3, 5, 9, 15, 25, 45, 75, 225.

Every factor of a number is a subpart of its prime factorisation: a product of some of its primes. Listing all subparts, together with 1, gives all its factors.

• Primes used | Subparts
• One | 3, 5
• Two | 3 × 3 = 9, 5 × 5 = 25, 3 × 5 = 15
• Three | 3 × 3 × 5 = 45, 3 × 5 × 5 = 75
• Four | 3 × 3 × 5 × 5 = 225

Notes

Every factor of a number is a subpart of its prime factorisation: a product of some of its primes. Listing all subparts, together with 1, gives all its factors.

Check yourself

840 = 2 × 2 × 2 × 3 × 5 × 7. Which of these is not a factor of 840?

150 = 2 × 3 × 5 × 5. The factor 25 is the subpart 5 × 5. The beads left over are 2 × 3. What is the partner factor, so that 25 × ? = 150?

Answer: 6

The leftover primes are 2 × 3 = 6, and 25 × 6 = 150.

Priya says: “18 is a factor of 36, because 18 = 2 × 3 × 3 and 36 = 2 × 2 × 3 × 3.” Is she right?

How many different factors does 36 have? Remember 36 = 2 × 2 × 3 × 3, and include 1 and 36.

Answer: 9

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36: nine of them.

• 3 × 3 × 3 — correct. Yes! 840 has only one 3, so there is no way to take three 3s as a subpart.
• 2 × 2 × 2. It is a factor: 840 has three 2s, so 2 × 2 × 2 = 8 is a subpart.
• 2 × 7. It is a factor: 2 × 7 = 14 is a subpart of the primes of 840.
• Yes: 2 × 3 × 3 is a subpart of 2 × 2 × 3 × 3. — correct. Yes! Pick one 2 and both 3s from the primes of 36. The leftover 2 is the partner: 36 = 18 × 2.
• No: 36 has two 2s but 18 has only one.. A subpart may use fewer primes than the number has. 18 needs only one 2, and 36 has two.
• No: 18 is not a prime.. Factors need not be primes. Any product of some of the primes is a factor.