Question

When positive integer N is divided by 167, the remainder is 35, and when positive integer K is divided by 167, the remainder is 17. What is the remainder when 2N+K is divided by 167?

Answer 1

$N\equiv35\mod167\implies N=167n_1+35$
$K\equiv17\mod167\implies K=167n_2+17$

for some integers $n_1,n_2$. So we have

$2N+K=2(167n_1+35)+(167n_2+17)$
$=167(2n_1+n_2)+87$
$=167n+87$

(where $n$ is some integer)

$\implies 2N+K\equiv87\mod167$

i.e. the remainder upon dividing $2N+K$ by 167 is 87.