Let a be any odd positive integer

Question

Let a be any odd positive integer and n be an integer greater than 5 .What is the smallest possible integer N such that $$a^N $$ is congruent to 1 modulo 2^n

Answer 1

Hint: This is $\mathrm{ord}_{2^n}(a)$. We know that

$$\mathrm{ord}_{2^n}(a)|\varphi(2^n)=2^{n-1},$$

so this number must be a power of $2$. Now see if you can apply lifting the exponent lemma to this to find the number of times $2$ divides $a^{2^k}-1$ for a given $k$.