Using prime factorisation

The teacher asks: are 56 and 63 co-prime? Anshu writes 56 = 14 × 4 and 63 = 21 × 3, sees no common factor, and says yes. But Guna writes 56 = 7 × 8 and 63 = 9 × 7, and 7 is a factor of both! So they are not co-prime. Anshu's way showed only some of the factors, not all of them.

Prime factorisation gives a sure check, because it shows every prime hidden inside a number. 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7. The prime 7 is in both, so they are not co-prime. Now 40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11 have no prime factor in common, so 40 and 231 are co-prime. If two numbers share no prime factor, they are co-prime.

Prime factorisation also tells us if one number is divisible by another. A number divides another only when it is hidden inside it. Is 168 divisible by 12? 168 = 2 × 2 × 2 × 3 × 7 and 12 = 2 × 2 × 3. The whole factorisation of 12 is included in that of 168: 168 = (2 × 2 × 3) × 2 × 7 = 12 × 14. So yes.

Is 75 divisible by 21? 75 = 3 × 5 × 5 and 21 = 3 × 7. 21 needs a 7, and 75 has none, so no. Is 42 divisible by 12? 42 = 2 × 3 × 7 and 12 = 2 × 2 × 3. 12 needs two 2s, but 42 has only one, so no. Every prime must be there as many times as it is needed.