A Multiplication Based Logic Puzzle

928 Halloween Cat

This cat has arrived just in time for Halloween. Find the factors that go with the clues in the grid to make this Halloween Cat puzzle a multiplication table:

Print the puzzles or type the solution on this excel file: 12 factors 923-931

Now let me tell you a little about the number 928.

It is the sum of four consecutive prime numbers:
227 + 229 + 233 + 239 = 928
and the sum of eight other consecutive prime numbers:
101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 928

It is the sum of two squares:
28² + 12² = 928
That means 928 is the hypotenuse of a Pythagorean triple:
640-672-928 which is 28² – 12², 2(28)(12), 28² + 12²

Here’s how 928 looks in a few other bases:
It’s 565 in BASE 13, because 5(169) + 6(13) + 5(1) = 928.
It’s 4A4 in BASE 14 (A is 10 base 10), because 4(196) + 10(14) + 4(1) = 928.
It’s TT in BASE 31 (T is 29 base 10), because 29(31) + 29(1) = 29(32) = 928.
It’s T0 in BASE 32, because 29(32) = 928.

  • 928 is a composite number.
  • Prime factorization: 928 = 2 × 2 × 2 × 2 × 2 × 29, which can be written 732 = 2⁵ × 29
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 928 has exactly 12 factors.
  • Factors of 928: 1, 2, 4, 8, 16, 29, 32, 58, 116, 232, 464, 928
  • Factor pairs: 928 = 1 × 928, 2 × 464, 4 × 232, 8 × 116, 16 × 58, or 29 × 32
  • Taking the factor pair with the largest square number factor, we get √928 = (√16)(√58) = 4√58 ≈ 30.463092423.

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