A Multiplication Based Logic Puzzle

127 and Level 6

127 is a prime number. 127 = 1 x 127. Its only factors are 1 and 127. Prime factorization: none.

How do we know that 127 is a prime number? The square root of 127 is an irrational number approximately equal to 11.27. If 127 were not a prime number, then it would be divisible by at least one prime number less than or equal to 11.27. Since 127 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 127 is a prime number.

Thinking process using divisibility tricks to prove that 127 is not divisible by 2, 3, 5, 7, or 11:

  • 127 is not even, so 127 is not divisible by 2.
  • 1 + 2 + 7 = 10 which is not divisible by 3, so 127 is not divisible by 3.
  • The last digit of 127 is not 5 or 0, so 127 is not divisible by 5.
  • The divisibility trick for 7 requires us to split 127 into 12 and 7. We double 7 and find the difference between the double and 12. We get (2 x 7) – 12  = 14 – 12 = 2. Since 2 is not divisible by 7, 127 is also not divisible by 7.
  • To perform the divisibility trick for 11 we look at the digits another way: Digits that are in an odd position are in regular type and the digit in an even position is in bold type: 127. We find the sum of the numbers typed in regular type (1 + 7  = 8) and the sum of the number(s) typed in bold type (2) and subtract the smaller sum from the larger sum (8 – 2 = 6). If the difference were a multiple of 11, than the original number would also be a  multiple of 11 (and divisible by 11). Since 6 is not a multiple of 11, we know 127 is not divisible by 11.

Therefore the only factor pair that exists for 127 is 1 x 127, and it is a prime number.

127 is never a clue in the FIND THE FACTORS puzzles.

Spotting the logic needed to begin this Level 6 puzzle should be easier than usual. If you have completed Level 5 puzzles, but have never before attempted a Level 6 puzzle, try this one !

2014-20 Level 6

Excel file of this week’s puzzles and last week’s factors: 12 Factors 2014-05-19

2014-20 Level 6 Logic

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Comments on: "127 and Level 6" (2)

  1. Grr! Took me 32 minutes! I kept making mistakes.

    Like

  2. ivasallay said:

    Sticking with it is one reason why you’re so grreat!

    Like

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