# Thread: x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)

1. ## x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)

Show that x^n ≡ 3 (mod 4) only if x ≡ n ≡ 1 (mod 2)

2. You mean: "if and only if" ?

Anyway: look at resctlasses modulo 4

That is: $\displaystyle \mathbb{Z}/4\mathbb{Z} = \left\{0,1,2,3\right\}$

Let $\displaystyle x\equiv 0$ mod 4. Then $\displaystyle x^n\equiv 0$ mod 4
Let $\displaystyle x\equiv 1$ mod 4. Then $\displaystyle x^n\equiv 1$ mod 4
Let $\displaystyle x\equiv 2$ mod 4. Then $\displaystyle x^n\equiv 0$ mod 4 for $\displaystyle n\geq 2$.

Let $\displaystyle x\equiv 3$ mod 4. Then $\displaystyle x^n\equiv 1$ mod 4 for even n. And $\displaystyle x^n\equiv 3$ mod 4 for odd n.

Hence the conclusion follows.

, mod