2016 AIME I Problems/Problem 12

Problem

Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.

Solution

We claim $m = 132$ . Note $132^2 - 132 + 11 = 11^3 \cdot 13$ .

Suppose $m < 132$ and $m^2 - m + 11 = pqrs$ for primes $p, q, r, s$ . We easily verify that $p, q, r, s \ge 11$ using quadratic residue argument. But $p, q, r, s = 11$ yields no integer solution for $m$ . Thus $pqrs \ge 11^3 \cdot 13$ . But this requires $m \ge 132$ from solving the quadratic, contradiction. Hence $m = \fbox{132}$ .

2016 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2016 AIME I Problems/Problem 12

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