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 1

$m(m-1)$ is the product of two consecutive integers, so it is always even. Thus $m(m-1)+11$ is odd and never divisible by $2$ . Thus any prime $p$ that divides $m^2-m+11$ must divide $4m^2-4m+44=(2m-1)^2+43$ . We see that $(2m-1)^2\equiv -43\pmod{p}$ . We can verify that $-43$ is not a perfect square mod $p$ for each of $p=3,5,7$ . Therefore, all prime factors of $m^2-m+11$ are $\ge 11$ .

Let $m^2 - m + 11 = pqrs$ for primes $11\le p \le q \le r \le s$ . From here, we could go a few different ways:

Solution 1a

Suppose $p=11$ ; then $m^2-m+11=11qrs$ . Reducing modulo 11, we get $m\equiv 1,0 \pmod{11}$ so $k(11k\pm 1)+1 = qrs$ .

Suppose $q=11$ . Then we must have $11k^2\pm k + 1 = 11rs$ , which leads to $k\equiv \mp 1 \pmod{11}$ , i.e., $k\in \{1,10,12,21,23,\ldots\}$ .

$k=1$ leads to $rs=1$ (impossible)! Then $k=10$ leads to $rs=101$ , a prime (impossible). Finally, for $k=12$ we get $rs=143=11\cdot 13$ .

Thus our answer is $m=11k= \boxed{132}$ .

Solution 1b

Let $m^2 - m + 11 = pqrs$ for primes $p, q, r, s\ge11$ . If $p, q, r, s = 11$ , then $m^2-m+11=11^4$ . We can multiply this by $4$ and complete the square to find $(2m-1)^2=4\cdot 11^4-43$ . But $(2\cdot 11^2-1)^2=4\cdot 11^4-4\cdot 11^2+1 <4\cdot 11^4-43<(2\cdot 11^2)^2,$ hence we have pinned a perfect square $(2m-1)^2=4\cdot 11^4-43$ strictly between two consecutive perfect squares, a contradiction. Hence $pqrs \ge 11^3 \cdot 13$ . Thus $m^2-m+11\ge 11^3\cdot 13$ , or $(m-132)(m+131)\ge0$ . From the inequality, we see that $m \ge 132$ . $132^2 - 132 + 11 = 11^3 \cdot 13$ , so $m = \boxed{132}$ and we are done.

Solution 2

First, we can show that $m^2 - m + 11 \not |$ $2,3,5,7$ . This can be done by just testing all residue classes.

For example, we can test $m \equiv 0 \mod 2$ or $m \equiv 1 \mod 2$ to show that $m^2 - m + 11$ is not divisible by 2.

Case 1: m = 2k

Case 2: m = 2k+1

Now, we can test $m^2 - m + 11 = 11^4$ , which fails, so we test $m^2 - m + 11 = 11^3 \cdot 13$ , and we get m = $132$ .

-AlexLikeMath

Video Solution

https://youtu.be/KRleD8iDRhI

~MathProblemSolvingSkills.com

2016 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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