A Multiplication Based Logic Puzzle

If you’ve always wanted to know the multiplication facts better, there is hope for you to do that in this bottle! Just write the numbers from 1 to 10 in the top row and also in the first column in an order that makes those factors and the given clues act like a multiplication table.

Print the puzzles or type the solution on this excel file: 10-factors-853-863

855 is the hypotenuse of Pythagorean triple 513-684-855 which is (3, 4, 5) times 171.

From Stetson.edu I learned that 855 can be expressed as sum of five consecutive squares (11² + 12² + 13² + 14² + 15² = 855) and the sum of two consecutive cubes (7³ + 8³ = 855). 855 is the smallest number that can make such a claim.

  • 855 is a composite number.
  • Prime factorization: 855 = 3 × 3 × 5 × 19, which can be written 855 = 3² × 5 × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 855 has exactly 12 factors.
  • Factors of 855: 1, 3, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855
  • Factor pairs: 855 = 1 × 855, 3 × 285, 5 × 171, 9 × 95, 15 × 57, or 19 × 45,
  • Taking the factor pair with the largest square number factor, we get √855 = (√9)(√95) = 3√95 ≈ 29.240383

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