Today I factor the number 500. How many factors does it have? Each number between 401 and 500 has at least 2 factors, but no more than 24 factors.

What if we had a horse race between the number of factors? Click on the graphic below to see a gif of the numbers racing against each other. Before you click, pick your pony. Will 2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, or 24 be the number of factors of more integers between 401 and 500 than any other number? Click on the graphic to find out!

Did you see the lead change a couple of times? How did your pony do? Which pony will you choose in the 501 to 600 race?

Remarkably, only 37 of these one hundred numbers have reducible square roots. That’s only 37%, which is significantly lower than in the 40% or 39% of previous hundreds as this graphic illustrates:

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- 500 is a composite number.
- Prime factorization: 500 = 2 x 2 x 5 x 5 x 5, which can be written 500 = (2^2) x (5^3)
- The exponents in the prime factorization are 2 and 3. Adding one to each and multiplying we get (2 + 1)(3 + 1) = 3 x 4 = 12. Therefore 500 has exactly 12 factors.
- Factors of 500: 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500
- Factor pairs: 500 = 1 x 500, 2 x 250, 4 x 125, 5 x 100, 10 x 50, or 20 x 25
- Taking the factor pair with the largest square number factor, we get √500 = (√100)(√5) = 10√5 ≈ 22.36067977

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If you didn’t want to click, you can still see the horse race below, but the numbers from 401 to 500 will be much clearer if you click.

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