FORM GR1268 Question 26

Question:
\[\begin{align}
3x&\equiv 5\pmod{11}\\
2y&\equiv 7\pmod{11}
\end{align}\]
If \(x\) and \(y\) are integers that satisfy the congruences above, then \(x+y\) is congruent module \(11\) to which of the following?

(A) \(1\)     (B) \(3\)     (C) \(5\)     (D) \(7\)     (E) \(9\)

Answer:
(D)

Answer Key:
Multiply the first equation by \(2\) to obtain \(6x\equiv10\pmod{11}\).
Multiply the second equation by \(3\) to obtain \(6y\equiv21\equiv10\pmod{11}\).
Add the two equations to obtain \(6(x+y)\equiv20\equiv9\pmod{11}\).

Compare this equation to
\(6\cdot1=6\equiv6\pmod{11}\)
\(6\cdot3=18\equiv7\pmod{11}\)
\(6\cdot5=30\equiv8\pmod{11}\)
\(6\cdot7=42\equiv9\pmod{11}\)
\(6\cdot9=54\equiv10\pmod{11}\)