A Multiplication Based Logic Puzzle

To find √486, I like to first see if 486 can be evenly divided by square numbers 4 or 9 because about 82% of numbers that can have their square roots reduced are divisible by 4 and/or by 9. It is also very easy to tell if a number can be evenly divided by 4 or by 9.

86 is not divisible by 4, so 486 is not divisible by 4.

However, 4 + 8 + 6 = 18, a multiple of 9 so 486 is divisible by 9, and now we know for sure that its square root can be reduced. 486 ÷ 9 = 54, another multiple of 9 so we get this two layer cake when we divide by 9 twice.

486 cake

Now take the square root of everything on the outside of the cake, and we get √486 = (√9)(√9)(√6) = 9√6.

Here is today’s Find the Factors puzzle:

486 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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  • 486 is a composite number.
  • Prime factorization: 486 = 2 x 3 x 3 x 3 x 3 x 3, which can be written 486 = 2 x (3^5)
  • The exponents in the prime factorization are 1 and 5. Adding one to each and multiplying we get (1 + 1)(5 + 1) = 2 x 6 = 12. Therefore 486 has exactly 12 factors.
  • Factors of 486: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486
  • Factor pairs: 486 = 1 x 486, 2 x 243, 3 x 162, 6 x 81, 9 x 54, or 18 x 27
  • Taking the factor pair with the largest square number factor, we get √486 = (√81)(√6) = 9√6 ≈ 22.04540

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486 Logic

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Comments on: "Finding √486 and Level 6" (2)

  1. 82% of numbers that can have their square roots reduced are divisible by 4 and/or by 9? Now that’s a fact one doesn’t come by every day! That rather boggles my mind.

    Liked by 1 person

    • ivasallay said:

      Hardly anyone is aware of this fact, too. Those who know it can have an advantage over those who don’t anytime a square root needs to be reduced.

      Liked by 1 person

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