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?

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anonymous · Answer

If $\dfrac{N}{167}$ gives a remainder of $35$, then $\dfrac{2N}{167}$ will have a remainder of $70$ (twice $35$). Then if $\dfrac{K}{167}$ has remainder $17$, adding $\dfrac{2N+K}{167}$ would add the remainders as well to get $\dfrac{2(35)+17}{167}=\dfrac{87}{167}$. Had the sum of the remainders been larger than $167$, you would have to make a slight adjustment. For example, if you had a total remainder of $\dfrac{168}{167}$, then the numerator is $1$ greater than $167$, which means your remainder would have been $1$. Notice that any remainders that differ by $167$ would actually be equivalent. You can learn more here if you so choose. https://en.wikipedia.org/wiki/Modular_arithmetic#Remainders