username42530853
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Solve the congruence problem x^15 ≡ 48 (mod 59).

x^15 ≡ 48 (mod 59) => x^3= ≡ 48^2 ≡ 3 (mod 59) ; why is this? How can we show that 48^2 ≡ 3 (mod 59)?
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RDKGames
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(Original post by username42530853)
Solve the congruence problem x^15 ≡ 48 (mod 59).

x^15 ≡ 48 (mod 59) => x^3= ≡ 48^2 ≡ 3 (mod 59) ; why is this? How can we show that 48^2 ≡ 3 (mod 59)?
One simple way is to note that

48^2 = (59-11)^2 = 59^2 - 2\cdot 11 \cdot 59 + 11^2

So

48^2 mod 59 is the same as 11^2 mod 59

and this is much easier to work out as 3 mod 59.
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