# Puzzle #7: Odds And Ends

Here's an interesting question for you. Which numbers have an *odd* number of positive integer divisors? If you're not sure what I mean by that, consider the number 12.

Twelve has the following positive integer divisors: 1, 2, 3, 4, 6, and 12. That's because 12 can be divided evenly by each one of those numbers. How many positive integer divisors does it have?

Six. That's an even number. Let's try another.

The positive integer divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24. That's eight divisors, which is *also* even. Hmm. It seems like it will always be an even number of divisors, because they come in pairs. Right? 1 and 24, 2 and 12, 3 and 8, 4 and 6.

So how can you have an *odd* number of positive integer divisors? I guess the only way would be if one of your *pairs* is actually one number multiplied by itself, like this:

36 = 1 x 36

36 = 2 x 18

36 = 3 x 12

36 = 4 x 9**36 = 6 x 6**

Ah ha! That's it! If a number is a *perfect square*, it has an odd number of divisors! Thirty-six, for example, has nine positive integer divisors!

Well, enough of that. Now it's time for you to decode your secret message. This one is *very short!*

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