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.

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:

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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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.

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.