Scharezade + Math

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

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?


  1. 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.

  2. 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.

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