Prove by Mathematical Induction $3^{2n}\equiv 1\pmod 4$ for every natural number n.

Question

Prove by Mathematical Induction $3^{2n}\equiv 1 \pmod 4$ for every natural number n.

Answer 1

For $n=1$ $3^2\equiv 1 \pmod 4$ Now suppose that $$3^{2n}\equiv 1\pmod 4$$ and you have to demonstrate that $3^{2n+2}\equiv 1\pmod 4$ Indeed $3^{2n}\cdot 3^2\equiv 1\pmod 4$

Answer 2

HINT: we have $3^{2(n+1)}=3^{2n}\cdot 3^2$

Answer 3

Base case : $9\equiv 1\ (\ mod\ 4\ )$ is true.

Suppose $9^n\equiv 1\ (\ mod\ 4\ )$

Then $9^{n+1}=9\times 9^n\equiv 9\equiv 1\ (\ mod\ 4\ )$

Answer 4

Actually, $3^{2n}\equiv 1\bmod 8$.

Indeed, by the binomial theorem, $3^{2n} = 9^n = (8+1)^n = 8a + 1$.