A Multiplication Based Logic Puzzle

396 is a multiple of 4, but not of 8, so just like 12, 20, 28, and 36, it is a leg of a primitive Pythagorean triple that is included in this infinite sequence of primitive triples (12513), (202129), (284553), (367785) . . . , which I’ve illustrated below:

20-21-29 What Pythagorean Triple Comes Next

Because they are Pythagorean triples, we know that 12² + 5² = 13², 20² + 21² = 29², 28² + 45² = 53², 36² + 77² = 85², and so forth.

What Pythagorean Triple Comes Next? Try to figure it out yourself, then scroll down a little bit to see if you are correct. In the meantime, let me tell you a little bit about the number 396:

  • 396 is a composite number.
  • Prime factorization: 396 = 2 x 2 x 3 x 3 x 11, which can be written 396 = (2^2) x (3^2) x 11
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 396 has exactly 18 factors.
  • Factors of 396: 1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396
  • Factor pairs: 396 = 1 x 396, 2 x 198, 3 x 132, 4 x 99, 6 x 66, 9 x 44, 11 x 36, 12 x 33 or 18 x 22
  • Taking the factor pair with the largest square number factor, we get √396 = (√11)(√36) = 6√11 ≈ 19.8997

The next primitive Pythagorean triple in the sequence can be illustrated like this:

11-117-125

Let me tell you about five Pythagorean triples in which 396 is one of the legs:

  • The answer to which Pythagorean triple comes next was (44117125), and is illustrated above. If we multiply that triple by 9, we get (396-1053-1125).
  • Because 396 equals 36 x 11, another triple can be found by multiplying the previous primitive in the sequence (367785) by 11 to get (396-847-935).
  • If we multiply the first triple in the sequence (12513by 33, we get (396-165-429).
  • The 16th primitive triple in the sequence is (13210851093). If we multiply it by 3 we get (396-3255-3279).
  • The 49th Pythagorean triple in our sequence of primitive triples above has a short leg that could be illustrated with 396 yellow squares. That primitive Pythagorean triple is (39697979805).

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