Every factor is a subpart

Once you have a number's prime factorisation, you have all its factors too — because every factor is built from a SUBPART of that factorisation.

Take 840, whose prime factorisation is 2 × 2 × 2 × 3 × 5 × 7. Is 2 × 2 × 7 = 28 a factor of 840? Reorder the primes — which does not change the product — into 840 = (2 × 2 × 7) × (2 × 3 × 5).

That reads 840 = 28 × 30. So yes, 28 is a factor of 840, and it must be multiplied by 30 to get there. The reordering did the whole job.

Now test others by the same idea. Is 2 × 7 = 14 a factor? Yes — 840 has at least one 2 and one 7 to spare. Is 2 × 2 × 2 a factor? Yes — 840 has exactly three 2s. Is 3 × 3 × 3 a factor? NO — 840's factorisation contains only ONE 3, so three of them are not available.

That is the test: a product is a factor exactly when every prime in it appears in the number's factorisation at least as many times.

Turn it round and you can LIST all the factors. Take 225 = 5 × 5 × 3 × 3. Build the subparts systematically. Single primes: 3, 5. Two primes: 3 × 3 = 9, 5 × 5 = 25, 3 × 5 = 15.

Three primes: 3 × 3 × 5 = 45, 3 × 5 × 5 = 75. Four primes: 3 × 3 × 5 × 5 = 225. Add 1, which is the empty subpart, and the factors of 225 are 1, 3, 5, 9, 15, 25, 45, 75, 225 — nine of them, with nothing missed.