A Multiplication Based Logic Puzzle

  • 693 is a composite number.
  • Prime factorization: 693 = 3 x 3 x 7 x 11, which can be written 693 = (3^2) x 7 x 11
  • 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 x 2 x 2 = 12. Therefore 693 has exactly 12 factors.
  • Factors of 693: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693
  • Factor pairs: 693 = 1 x 693, 3 x 231, 7 x 99, 9 x 77, 11 x 63, or 21 x 33
  • Taking the factor pair with the largest square number factor, we get √693 = (√9)(√77) = 3√77 ≈ 26.324893

Some quick divisibility tricks applied to the number 693:

  1. Every counting number is divisible by 1
  2. 693 is not even so it isn’t divisible by 2
  3. Every digit of 693 is divisible by 3, so 693 is divisible by 3
  4. Since it isn’t divisible by 2, it isn’t divisible by 4
  5. 693 doesn’t end in a 5 or 0, so it’s not divisible by 5
  6. 693 is divisible by 3 but not by 2 so it isn’t divisible by 6
  7. 69 – 2(3) = 63, a multiple of 7 so 693 is divisible by 7
  8. Since it isn’t divisible by 2 or 4, it can’t be divisible by 8
  9. 6 + 9 + 3 = 18, a multiple of 9 so 693 is divisible by 9
  10. Since the last digit of 693 isn’t 0, it is not divisible by 10
  11. 69 + 3 = 0, so 693 is divisible by 11

The divisibility tricks that worked on the number 693 are quite easy to see on the outside of this factor cake.

Here is today’s factoring puzzle:

693 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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693 is a palindrome in several bases:

  • 1010110101 BASE 2; note 1(512) + 0(256) + 1(128) + 0(64) + 1(32) + 1(16) + 0(8) + 1(4) + 0(2) + 1(1) = 693
  • 3113 BASE 6; note 3(216) + 1(36) + 1(6) + 3(1) = 693
  • 414 BASE 13; note 4(169) + 1(13) + 4(1) = 693
  • 313 BASE 15; note 3(225) + 1(15) + 3(1) = 693

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693 Logic

Ricardo tweeted his work for this puzzle, too.

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