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Right triangle with a 23

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Find two positive integers that together with 23 are the lengths of a right triangle.

Hint: There’s a simple technique that, given any odd positive integer, allows you to figure out the other two integer sides of a right triangle in your head (or with pen and paper if the numbers get too large).  Find this technique.

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ronret45 Expert Asked on 1st August 2015 in Microsoft Interview Puzzles.
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    Best answer

    You can generate Pythagorean triples using Euclid’s formula:

    a = m^2 – n^2
    b = 2mn
    c = m^2 + n^2

    But the key thing to note here is a = m^2 – n^2 = (m – n) (m + n). So given any odd positive integer (except 1), we can force m – n = 1 and so a = m + n. For example for 23, we can make m = 12 and n = 11, and so a = (12 – 11) (12 + 11) = 23. Therefore the other 2 integers are b = 2mn = 2 * 12 * 11 = 264 and c = m^2 + n^2 = 12^2 + 11^2 = 265.

    In summary given a odd integer k, we can find the 2 other integers, b and c, such that k, b, c make a Pythogorean triangle:
    b = (k + 1) (k – 1) / 2
    c = ((m + 1)^2 + (m – 1)^2) / 4

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    Ralph Aldanese Scholar Answered on 4th March 2016.
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