Factors and common factors

Jumpy and Grumpy play Jump Jackpot. Grumpy puts a treasure on a number, say 24. Jumpy picks a jump size and jumps along the number line from 0, always the same size. With jump size 4 he lands on 4, 8, 12, 16, 20, 24, ... and gets the treasure! (One jump of 4 lands on 4, two jumps on 8, three jumps on 12: multiplying is just counting the jumps.)

Which jump sizes land on 24? The sizes 1, 2, 3, 4, 6, 8, 12 and 24 all do. These are exactly the numbers that divide 24 with nothing left over. They are called the factors (or divisors) of 24. A jump size like 5 misses: 5, 10, 15, 20, 25 jumps right over 24. Jump size 7 misses too: 7, 14, 21, then 28. Three 7s fit into 24, with 3 left over.

Then Grumpy makes it harder: two treasures, on 14 and 36, and Jumpy must keep the same jump size. He picks 7 and lands on 7, 14, 21, 28, 35, 42, ... He gets 14 but misses 36. The factors of 14 are 1, 2, 7, 14. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. Only 1 and 2 are in both lists, so only jump sizes 1 and 2 reach both treasures.

Numbers that are factors of two numbers at once are called their common factors. The jump sizes that reach both treasures are the common factors of the two numbers. For treasures on 28 and 70: the factors of 28 are 1, 2, 4, 7, 14, 28 and the factors of 70 are 1, 2, 5, 7, 10, 14, 35, 70, so the common factors are 1, 2, 7 and 14.