It is given 13 integers $\displaystyle c_1,c_2,...,c_{13}$ (some of them may be the same). Use pigeonhole principle to prove that there exist i and j with $\displaystyle 0<i<j<=13$ such that

$\displaystyle c_{i+1}+c_{i+2}+...+c_j$ is divisible by 13

for example, $\displaystyle c_4+c_5+c_6+c_7$ is divisible by 13.

(Hint:consider the following 13 integers

$\displaystyle n_1=c_1$

$\displaystyle n_2=c_1+c_2$

.

.

.

$\displaystyle n_{13}=c_1+c_2+...+c_{13}$

and their remainder when divided by 13)