Show that there are two distinct positive integers such that: $1394|2^a-2^b$

Question

Show that there are two distinct positive integers such that: $1394|2^a-2^b$

I'm sure pigeon hole principle applies here,but don't recognize holes.Another problem statement is: show that there are two positive integers $a,b$ such that: $$2^a\equiv 2^b\pmod {1394}$$
Of course we have $1394$ cases for division mod $1394$,but what are the pigeons?

Answer 1

Hint:

Consider $2^i \pmod{1394}$ for $1 \leq i \leq \color{blue}{1395}$

Answer 2

Note $1394=2 \times 17 \times 41$.

Since $\varphi(17 \times 41) = 16\times 40 = 640$,

then $2^{640} \equiv 1 \pmod{697}$.

Hence $2^{641} \equiv 2 \pmod{1394}$.

In other words, $1394 \mid 2^{641}-2^1$.

Note. Actually, we can do better. Checking the divisors of $640$, we find that $2^{40} \equiv 1 \pmod{697}$. So $1394 \mid 2^{41}-2^1$.