A Multiplication Based Logic Puzzle

198 and Level 6

  • 198 is a composite number.
  • Prime factorization: 198 = 2 × 3 × 3 × 11, which can be written 198 = 2 × 3² × 11
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 198 has exactly 12 factors.
  • Factors of 198: 1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198
  • Factor pairs: 198 = 1 × 198, 2 × 99, 3 × 66, 6 × 33, 9 × 22, or 11 × 18
  • Taking the factor pair with the largest square number factor, we get √198 = (√9)(√22) = 3√22 ≈ 14.0712

Since 1 + 9 + 8 = 18, a multiple of 9, 198 is divisible by 9. Here is one of several possible factor trees for 198:

198 factor tree

2014-31 Level 6

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-04

The common factors of 20 and 40 are 4, 5, and 10. Since 14 and either 8, 10, or 6 will use both 2’s, 10 cannot be the CF. If we choose 4 as the common factor, then 5 and 10 would be placed in the top row followed by 3, the factor of 15. However, then we wouldn’t be able to use either 3 or 6 as the common factor of 24 and 18 because 3 would already be in the top row and 4 would already be in the 1st column.  Therefore, the common factor of 20 and 40 has to be 5.

2014-31 Level 6 Logic

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