Question:
The remainder when \((556)^{40}\) is divided by 7 is ________.
The remainder when \((556)^{40}\) is divided by 7 is ________.
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Reduce 556 modulo 7 first, then use the fact that \(3^6\equiv 1 \pmod 7\).
Updated On: Nov 12, 2026
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Correct Answer: 4
Solution and Explanation
\(556=553+3=7\times 79+3\), so \(556\equiv 3 \pmod 7\).
Hence \(556^{40}\equiv 3^{40}\pmod 7\).
\(3^2=9\equiv 2\), \(3^3\equiv 6\equiv -1\), so \(3^6\equiv 1\).
\(40=6\times 6+4\), therefore \(3^{40}\equiv 3^4=81\). Since \(81=77+4\), the remainder is 4.
Answer: 4
Hence \(556^{40}\equiv 3^{40}\pmod 7\).
\(3^2=9\equiv 2\), \(3^3\equiv 6\equiv -1\), so \(3^6\equiv 1\).
\(40=6\times 6+4\), therefore \(3^{40}\equiv 3^4=81\). Since \(81=77+4\), the remainder is 4.
Answer: 4
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