Finding mystery integers?

Question

1) Find an integer, x, greater than 4 where the sum of x consecutive integers is divisible by x

2) Find an integer, y, greater than 4 where the sum of y consecutive integers is NOT divisible by y

Answer 1

1)  x = 5, so x > 4.

5 + 6 + 7 + 8 + 9 = 35

35 is divisible by 5.

6 + 7 + 8 + 9 + 10 = 40

40 is divisible by 5.

In general for any integer n,

n + (n + 1) + (n + 2) + (n + 3) + (n + 4) =

5n + 1 + 2 + 3 + 4 =

5n + 10

(5n + 10) is divisible by 5 for any n.

2)  y = 6, so y > 4.

5 + 6 + 7 + 8 + 9 + 10 = 45

45 is not divisible by 6.

6 + 7 + 8 + 9 + 10 + 11 = 51

51 is not divisible by 6.

In general, for any integer n,

n + (n + 1) + (n + 2) + (n + 3) + (n + 4) + (n + 5) =

6n + 1 + 2 + 3 + 4 + 5 =

6n + 15

6n + 15 is not divisible by 6 for any n.

Hope this helps you!