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Harvard-MIT Math Tournament

Algebra Difficulty 4.6 AIME Prove it United States

Problem:
Given that $7,999,999,999$ has at most two prime factors, find its largest prime factor.

Solution

Solution:
$7,999,999,999 = 8 \cdot 10^{9} - 1 = 2000^{3} - 1 = (2000 - 1)\left(2000^{2} + 2000 + 1\right)$ , so $\left(2000^{2} + 2000 + 1\right) = 4,002,001$ is its largest prime factor.

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