GeekDad Puzzle of the Week – Squared Roots

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Numbers are, for the most part, made up of other numbers. Prime numbers (2, 3, 7, etc.) are simple, in that they make up themselves — they have no non-trivial factors, and can only be divided by the natural numbers 1 and themselves. Compound numbers, on the other hand, may have entirely different roots.

Some compound numbers, like 6 or 21, are still relatively simple — they are the product of two primes. Other compound numbers, like 144 or 1183, can be divided by squared numbers — 144 can be divided by 4, 9, 16, and 144, and 1183 can be divided by 169 (or 132.) Both of these numbers definitely have factors of the form n2, or (to coin a math pun) “squared roots.”

A collection of square roots from around the interwebs.
A collection of squared roots from around the interwebs.

This week’s GeekDad Puzzle of the Week deals with “squared roots,” or more specifically, numbers without any squared roots.

The question: How many of the numbers under 10 million do not have squared roots; that is, how many numbers under 107 cannot be evenly divided by a squared number?

As always, please send your response in to GeekDad Central for your chance at this week’s $50 Gift Certificate from ThinkGeek. The winner will be drawn at random from among the correct and reasonably correct (or at least well supported) responses sent in between now and Saturday, 22 Mar 2014.

Good luck, and happy puzzling!

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