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how can I find the last two digits of 91^92 ?
how can I find the last two digits of 91^92 ?
Euler's theorem -- phi(100) = 40, so we can reduce 92 mod 40 to get 91^92 ≡ 91^12 (mod 100).
The binary expansion of 12 is 1100_2, so we can calculate quickly by repeated squaring:
91^2 ≡ (-9)^2 ≡ 81 (mod 100)
91^4 ≡ 81^2 ≡ 61 (mod 100)
91^8 ≡ 61^2 ≡ 21 (mod 100)
91^12 = 91^(4+8) ≡ 61 * 21 ≡ 81 (mod 100)
The answer is 81.
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