1413 and Level 4

You find a rhythm as you solve this level 4 puzzle. The logic is quite easy for most of it, but there is at least one place that will require you to think things through before proceeding.

Now I’ll share a few facts about the puzzle number, 1413:

  • 1413 is a composite number.
  • Prime factorization: 1413 = 3 × 3 × 157, which can be written 1413 = 3² × 157
  • 1413 has at least one exponent greater than 1 in its prime factorization so √1413 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1413 = (√9)(√157) = 3√157
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1413 has exactly 6 factors.
  • The factors of 1413 are outlined with their factor pair partners in the graphic below.

1413 is the sum of two squares:
33² + 18² = 1413

That means that 1413 is the hypotenuse of a Pythagorean triple:
765-1188-1413 calculated from 33² – 18², 2(33)(18), 33² + 18².
It is also 9 times (85-132-157)

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