88 is divisible by 4 so 588 is also divisible by 4, and that means √588 can be reduced.

About 83% of the numbers that have reducible square roots are divisible by 4 and/or by 9, and it is so easy to tell if even a very long number is divisible by either of those numbers. It is also easier to divide a number by 4 or 9 than it is to divide by their square roots twice.

When I reduce a square root, I like to make a little cake and start by dividing by 100, 4, or 9 if any of those numbers are its factors. Here are the steps I used to make a cake for 588 with as many perfect squares on the outside of the cake as possible.

- 588 ÷ 4 = 147
- 147 is not divisible by 4 again, but 5 + 8 + 8 = 21 so 147 is divisible by 3, but not by 9.
- 147 ÷ 3 = 49 which is a perfect square, so I stop dividing and simply take the square roots of everything on the outside of the cake and multiply them together.

This is what my cake looks like:

And now for today’s Level 3 puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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- 588 is a composite number.
- Prime factorization: 588 = 2 x 2 x 3 x 7 x 7, which can be written 588 = (2^2) x 3 x (7^2)
- The exponents in the prime factorization are 2, 1 and 2. Adding one to each and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 x 2 x 3 = 18. Therefore 588 has exactly 18 factors.
- Factors of 588: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588
- Factor pairs: 588 = 1 x 588, 2 x 294, 3 x 196, 4 x 147, 6 x 98, 7 x 84, 12 x 49, 14 x 42 or 21 x 28
- Taking the factor pair with the largest square number factor, we get √588 = (√196)(√3) = 14√3 ≈ 24.248711

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**A Logical Approach to solve a FIND THE FACTORS puzzle**: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row). ** Because this is a level three puzzle**, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

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