A Multiplication Based Logic Puzzle

  • 696 is a composite number.
  • Prime factorization: 696 = 2 x 2 x 2 x 3 x 29, which can be written 696 = (2^3) x 3 x 29
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 696 has exactly 16 factors.
  • Factors of 696: 1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348, 696
  • Factor pairs: 696 = 1 x 696, 2 x 348, 3 x 232, 4 x 174, 6 x 116, 8 x 87, 12 x 58, or 24 x 29
  • Taking the factor pair with the largest square number factor, we get √696 = (√4)(√174) = 2√174 ≈ 26.38181.

Today’s puzzle is meant to look like a Christmas stocking or boot that can be filled with lots of little treasures.

696 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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What other facts did I find about the number 696?

696 is the sum of all the prime numbers from 71 to 103. Do you know what those eight prime numbers are?

696 is also the sum of consecutive odd numbers 347 and 349 which just happen to also be consecutive prime numbers.

Because 696 is a multiple of 29, it is the hypotenuse of Pythagorean triple 480-504-696. What is the greatest common factor of those three numbers?

696 is a palindrome in two different bases

  • 696 BASE 10; note that 6(100) + 9(10) + 6(1) = 696
  • OO BASE 28; note that O BASE 28 is equivalent to 24 in BASE 10, and  24(28) + 24(1) = 696

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696 Factors

 

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