HMMT February

Number theory Difficulty 4.7 AIME Find the answer

Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^{n}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have $7^{2048}-1=(7-1)(7+1)\left(7^{2}+1\right)\left(7^{4}+1\right) \cdots\left(7^{1024}+1\right)$ In the expansion, the eleven terms other than $7+1$ are divisible by 2 exactly once, as can be checked easily with modulo 4.

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