2016 AIME I Problems/Problem 12
Problem
Find the least positive integer such that
is a product of at least four not necessarily distinct primes.
Solution
We claim . Note
.
Suppose and
for primes
. We easily verify that
using quadratic residue argument. But
yields no integer solution for
. Thus
. But this requires
from solving the quadratic, contradiction. Hence
.
See Also
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 | ||
All AIME Problems and Solutions |
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