Without actually computing 12 explain how you know that 12

Without actually computing 12!, explain how you know that 12! + 11 is divisible by 11. Without actually computing any of the numbers below, explain how you know each of the numbers is composite. 7! + 2, 7! + 3, 7! + 4, 7! + 5, 7! + 6, 7! + 7 Without actually computing any numbers, explain how you could create a list of 12 consecutive numbers, all of which are composite.

Solution

since 7! is multiple of 7,6,5,4,3,2,

7!+2 here 7! ,2 divisible by 2, 7!+2 is a multiple of 2 so it is composite

7!+3 here 7! ,3 divisible by 3, 7!+3 is a multiple of 3 so it is composite

7!+4 here 7! ,4 divisible by 4, 7!+4 is a multiple of 4 so it is composite

7!+5 here 7! ,5 divisible by 5, 7!+5 is a multiple of 5 so it is composite

7!+6 here 7! ,6 divisible by 6, 7!+6 is a multiple of 6 so it is composite

7!+7 here 7! ,7 divisible by 7, 7!+7 is a multiple of 7so it is composite