Common factors

In your book, Maths Mela (Chapter 13, Animal Jumps), Mowgli's friends live along a numbered trail. The ant, the frog, the bird and the rabbit live at 4, 12, 14 and 50. Mowgli starts at 0 and jumps 2 steps each time: 2, 4, 6, 8, 10, 12, 14 … 50. He lands on all four homes, because 4, 12, 14 and 50 are all multiples of 2. We say 2 is a common factor of 4, 12, 14 and 50. In the same way, jumping 3 steps he meets friends at 9, 21, 39 and 57, so 3 is a common factor of those numbers.

A common factor of two or more numbers divides each of them exactly. It is a jump size that lands on every one of the numbers, starting from 0.

Let us find the common factors of 24 and 36. Can jumps of 2 reach both? Yes. Jumps of 3? Yes (3 × 8 = 24, 3 × 12 = 36). Jumps of 4? Yes (4 × 6 = 24, 4 × 9 = 36). Jumps of 6 and 12 work too, and so do jumps of 1, which reach every number. Jumps of 8 do not: they reach 24 but miss 36. So the common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12.

Some pairs share only one factor. The factors of 12 are 1, 2, 3, 4, 6, 12, and the factors of 13 are only 1 and 13. So 12 and 13 have just one common factor: 1. Since 1 and the number itself are always factors, 1 is a common factor of any numbers at all.