Co-prime numbers

Jumpy and Grumpy play the treasure game again, with treasures on two numbers. Jumpy gets them only if he reaches both with the same jump size. And there is a new rule: a jump size of 1 is not allowed. Where should Grumpy hide the treasures so that Jumpy can never reach both?

Treasures on 12 and 26 are not safe: with jump size 2, Jumpy reaches both. But treasures on 4 and 9 are safe! The factors of 4 are 1, 2, 4 and the factors of 9 are 1, 3, 9. Their only common factor is 1, and jump size 1 is not allowed.

Two numbers are called co-prime when they have no common factor other than 1. 4 and 9 are co-prime. 15 and 39 are not, because 3 is a factor of both (15 = 3 × 5 and 39 = 3 × 13). A safe pair of treasures is a co-prime pair. Co-prime is about a pair, not one number: the numbers themselves need not be prime. 63 and 20 are both composite, yet they are co-prime (63 = 3 × 3 × 7 and 20 = 2 × 2 × 5 share nothing but 1).

Anshu noticed something while playing the idli-vada game. For co-prime numbers, the first common multiple is their product: 3 and 5 first meet at 15 = 3 × 5, and 4 and 9 first meet at 36 = 4 × 9. For numbers that are not co-prime, the first common multiple is smaller than the product: 4 and 6 first meet at 12, which is less than 4 × 6 = 24.