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?

1 answer
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?

Answers

[tex]N\equiv35\mod167\implies N=167n_1+35[/tex][tex]K\equiv17\mod167\implies K=167n_2+17[/tex]for some integers [tex]n_1,n_2[/tex]. So we have[tex]2N+K=2(167n_1+35)+(167n_2+17)[/tex][tex]=167(2n_1+n_2)+87[/tex][tex]=167n+87[/tex](where [tex]n[/tex] is some integer)[tex]\implies 2N+K\equiv87\mod167[/tex]i.e. the remainder upon dividing [tex]2N+K[/tex] by 167 is 87.

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