Question

Prove that the sum of any 7 consecutive non- negative integers is divisible by 7 . Is this true for any positive integers instead of 7

Answer 1

Question #25992 Prove that the sum of any 7 consecutive non-negative integers is divisible by 7. Is this true for any positive integers instead of 7.

Solution Take any positive integer $n$, let be $k$ such that $n = k \mod 7$, then $n + i = k + i \mod 7$, thus the sum of any consecutive integers is $\sum_{i=0}^{6} n + i = 7k + 6(6 + 1)/2 = 0 \mod 7$. This is not true for all integers, really $3 + 4 = 1 \mod 2$, thus the sum of 2 consecutive integers is not divisible by 7.