1964 AHSME Problem 3

When a positive integer x is divided by a positive integer y, the quotient is u and the remainder is v, where u and v are integers. What is the remainder when x+2uy is divided by y?

\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2u \qquad \textbf{(C)}\ 3u \qquad \textbf{(D)}\ v \qquad \textbf{(E)}\ 2v

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.