A Multiplication Based Logic Puzzle

408 and Level 6

408 is divisible by 8 and so are 1408, 2408, 3408, 4408, . . . . . . and any other number whose last 3 digits are 408. This is true because if the last 3 digits of a number are divisible by 8, the entire number is also!

Also since 4 + 8 = 12, a multiple of 3, we know that 408 is divisible by 3.

408’s factors are listed below the puzzle.

408 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

  • 408 is a composite number.
  • Prime factorization: 408 = 2 x 2 x 2 x 3 x 17, which can be written 408 = (2^3) x 3 x 17
  • 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 408 has exactly 16 factors.
  • Factors of 408: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408
  • Factor pairs: 408 = 1 x 408, 2 x 204, 3 x 136, 4 x 102, 6 x 68, 8 x 51, 12 x 34, or 17 x 24
  • Taking the factor pair with the largest square number factor, we get √408 = (√4)(√102) = 2√102 ≈ 20.199

408 Logic

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