If x is the remainder when 3 61284 is divided by 5 and y is the remainder when 4 96 is divided by 6, then what is the value of (2x – y)?
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Step 1 — Find x (remainder of \(3^{61284}\) by 5):
Powers of 3 mod 5 cycle with length 4: \(3,4,2,1\). Since \(61284=4\times 15321\), the exponent is a multiple of 4, so the remainder is the 4th term:
\[3^{61284}\equiv 3^{4}\equiv 1\pmod 5\;\Rightarrow\;x=1\]
Step 2 — Find y (remainder of \(4^{96}\) by 6):
For every \(n\ge 1\), \(4^{n}\equiv 4\pmod 6\) (since \(4\equiv 1\pmod 3\) makes \(4^{n}-4\) divisible by both 2 and 3). Hence \(y=4\).
Step 3 — Evaluate \(2x-y\):
\[2(1)-4=-2\]
Therefore \(2x-y=\mathbf{-2}\).
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select the correct answer using the code given below:
