## Scharezade + Math

My favorite puzzler, Lee, over at primepuzzle sent this problem in. Have fun!

http://storynory.com/2010/08/09/the-1001-nights/

What are the lengths of the sides of the smallest similar triangle?

“A terrible sultan marries a new bride every night, and in the morning he executes her. Only Sherehezade, the greatest story-teller the world has known, has a chance to soften the heart of the man with a tyrannical grudge against all women.”

It probably won’t take you 1001 nights to solve this one but in honor of Sherehezade we present the following challenge:

Can you find a right triangle that has a hypotenuse of 1001 and sides that are whole numbers?

What are the lengths of the sides of the smallest similar triangle?

This entry was posted on Tuesday, June 19th, 2012 at 9:35 pm and is filed under problem.
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Huge hint:

Without using a computer program it would be very difficult to find two integers i and j such that i^2 + j^2 = 1001^2 = 1,002,001. Since most of us aren’t programmers, we’ve got to get smarter

Factor 1001 into its prime factors. The first few “primitive” “Pythagorean triples” are (3,4,5), 5,12,13), (8,15,17) etc. etc. The hope is that our triangle is *similar* to a “primitive” one. This would

mean the two legs have factors that are also factors of the hypotenuse.

Well … that was a flop.

The number 1001 factors to 7*11*13. There is a right triangle which has integral sides and a hypotenuse of 13, namely (5,12,13). So our triangle is simply the one that’s similar to it and 77 times it, namely 77*(5,12,13) or (385,924,1001).

We got lucky with 1001. It’s the third number in a “Pythagorean triple” that happens to be factorable into a multiple of the third number of a small Pythagorean Triple.

http://youtu.be/8fP_alomUaQ