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2025-2025-x-mod-17-

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Question Number 201418 by cortano12 last updated on 06/Dec/23
2025^(2025) = x (mod 17 )
20252025=x(mod17)
Answered by mr W last updated on 06/Dec/23
2025^(2025) (mod 17) =(119×17+2)^(2025) (mod 17) ≡2^(2025) (mod 17) =2×2^(4×506) (mod 17) =2×(17−1)^(506) (mod 17) ≡2×(−1)^(506) (mod 17) ≡2 (mod 17)
20252025(mod17)=(119×17+2)2025(mod17)≡22025(mod17)=2×24×506(mod17)=2×(17−1)506(mod17)≡2×(−1)506(mod17)≡2(mod17)
Answered by BaliramKumar last updated on 06/Dec/23
((2025^(2025) )/(17)) [ φ(17)= 16 ] (((17×119+2)^(16×126+9) )/(17)) (2^9 /(17)) = (((2^4 )^2 ×2^1 )/(17)) = (((16)^2 ×2)/(17)) (((17−1)^2 ×2)/(17)) = (((−1)^2 ×2)/(17)) ((1×2)/(17)) = (2/(17)) = 2
2025202517[ϕ(17)=16](17×119+2)16×126+9172917=(24)2×2117=(16)2×217(17−1)2×217=(−1)2×2171×217=217=2

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