Suppose $2^a \equiv 2^b\pmod{101}$. Is $a \equiv b \pmod {100}$ always true?
The first thing that came in my mind was Fermat's Little Theorem. WLOG $a\ge b$. Since $(101,2)=1$, dividing both sides by $2^b$ gives $$2^{a-b}\equiv 1\pmod {101}$$ Also, $$2^{100}\equiv 1\pmod {101} $$ by Fermat's Little Theorem.
How should I continue?
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