# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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A Square of Divisors. Topic: Number Theory. Level: AIME/Olympiad. May 8th, 2007

Problem: (1999 Canada – #3) Determine all positive integers with the property that . Here denotes the number of positive divisors of .

Solution: So, testing some small numbers yields as a solution. We claim that this is the only such solution.

Clearly, since is a square, we can use a variant of the usual prime decomposition and say that .

Furthermore, again since is a square, we know

is odd, so must be odd as well, i.e. is not one of the . Then we use the equation given to us to get

.

Note, however, that by Bernoulli’s Inequality (overkill, I know) we have for

with equality iff and . So

.

Since we want equality, we must have and for all . But since the primes are supposed to be distinct we can have exactly one prime so that is the only solution. QED.

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Comment: Another one of those problems that you kind of look at the result and you’re like huh, that’s interesting. But anyway, just throwing in some weak inequalities led to a pretty straightforward solution. As long as you know how to find the number of divisors of a positive integer it isn’t too much of a stretch to figure the rest out, though it make take some time to get in the right direction since the problem is quite open-ended.

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Practice Problem: (1999 Canada – #4) Suppose are eight distinct integers from . Show that there is an integer such that the equation has at least three different solutions. Also, find a specific set of distinct integers from such that the equation does not have three distinct solutions for any .

### 6 Responses to “A Square of Divisors. Topic: Number Theory. Level: AIME/Olympiad.”

1. Xuan Says:

This actually has nothing to do with your question, but i thought it was pretty interesting:

Isn’t nonexistence or existence only as an idea still a form of existence?

Which analogy is better?:

existence:nonexistence
1 : -1

existence:nonexistence
1 : 0

1 : 0.

Nonexistence is the absence of existence. And yes, the concept of nonexistence is a form of existence, but who claimed otherwise? Nonexistence does not have to describe itself.

3. blue_giraffe Says:

….

Hey! You should’ve gone to bed.

5. Evan Says:

Looking at this, it appears obvious that n = 1 is also a solution
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