A Multiplication Based Logic Puzzle

782 and Level 1

  • 782 is a composite number.
  • Prime factorization: 782 = 2 x 17 x 23
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 782 has exactly 8 factors.
  • Factors of 782: 1, 2, 17, 23, 34, 46, 391, 782
  • Factor pairs: 782 = 1 x 782, 2 x 391, 17 x 46, or 23 x 34
  • 782 has no square factors that allow its square root to be simplified. √782 ≈ 27.96426.

782-factor-pairs

Here’s today’s puzzle:

782-puzzle

 Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s some more information about the number 782:

Stetson.edu informs us that the sum of the factors of 782 is not only a perfect square number but also a perfect 4th power.

Specifically, 1 + 2 + 17 + 23 + 34 + 46 + 391 + 782 = 1296, which happens to be 6^4.

23 is one of the factors of 782, and here are three interesting number facts about the number 782 that involve the number 23:

  • 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 782. (That’s 23 consecutive numbers beginning with the number 23.)
  • 782 is the 23rd pentagonal number because 23(3⋅23 – 1)/2 = 782.
  • 782 is palindrome NN in BASE 33. (N is 23 base 10.) Note that 23(33) + 23(1) = 782

Because 17 is one of its factors, 782 is the hypotenuse of Pythagorean triple 368-690-782.

Thus, 368² + 690² = 782².

Also since 17 x 46 = 782, the greatest common factor of 368, 690 and 782 is 46.

782 is also the sum of three squares six different ways:

  • 27² + 7² + 2² = 782
  • 26² + 9² + 5² = 782
  • 25² + 11² + 6² = 782
  • 22² + 17² + 3² = 782
  • 19² + 15² + 14² = 782
  • 18² + 17² + 13² = 782

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782-factors

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