HMMT February

Number theory Difficulty 4.7 AIME Find the answer

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

Your answer

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

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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