Show that x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)
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You mean: "if and only if" ? Anyway: look at resctlasses modulo 4 That is: Let mod 4. Then mod 4 Let mod 4. Then mod 4 Let mod 4. Then mod 4 for . Let mod 4. Then mod 4 for even n. And mod 4 for odd n. Hence the conclusion follows.
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