Topic: Elementary Number Theory

Let x be an integer. let p be a prime. Suppose that x^2 == x mod p. Show that either x == 0 mod p, or x == 1 mod p

Note that we have

If x^2 ≡ x (mod p),

then x(x − 1) ≡ 0 (mod p).

Thus p | x(x − 1), so p | x or p | x − 1.

Hence the only solutions are x ≡ 0 (mod p) or x ≡ 1 (mod p).

Hence Proved

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