# 1964 AHSME Problems/Problem 3

## Problem

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 $y$?

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

## Solution 1

• We can solve this problem by elemetary modular arthmetic,

$x \equiv v\ (\textrm{mod}\ y)$ $=>$ $x+2uy \equiv v\ (\textrm{mod}\ y)$

~GEOMETRY-WIZARD

## Solution 2

By the definition of quotient and remainder, problem states that $x = uy + v$.

The problem asks to find the remainder of $x + 2uy$ when divided by $y$. Since $2uy$ is divisible by $y$, adding it to $x$ will not change the remainder. Therefore, the answer is $\boxed{\textbf{(D)}}$.

## Solution 3

If the statement is true for all values of $(x, y, u, v)$, then it must be true for a specific set of $(x, y, u, v)$.

If you let $x=43$ and $y = 8$, then the quotient is $u = 5$ and the remainder is $v = 3$. The problem asks what the remainder is when you divide $x + 2uy = 43 + 2 \cdot 5 \cdot 8 = 123$ by $8$. In this case, the remainder is $3$.

When you plug in $u=5$ and $v = 3$ into the answer choices, they become $0, 5, 10, 3, 6$, respectively. Therefore, the answer is $\boxed{\textbf{(D)}}$.