## A Multiplication Based Logic Puzzle

### 500 Pick Your Pony! Who’ll Win This Number of Factors Horse Race?

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?

While we’re on the subject of horse racing, you’ll get a chuckle from reading the mathemagical site’s place-your-bets.

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.

make animated gifs like this at MakeAGif

#### Comments on: "500 Pick Your Pony! Who’ll Win This Number of Factors Horse Race?" (6)

1. Congratulations! A milestone! I bet 2, so my pony lost.

Liked by 1 person

• ivasallay said:

Thank you! Your pony ran very fast and placed 3rd which is not quite the same as losing.

Liked by 1 person

2. Congratulations on reaching 500 posts!

I had figured on six winning out, myself.

Liked by 1 person

• ivasallay said:

Thank you, and also much thanks for participating and commenting!

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3. I’m rather out of circulation at the moment, but I couldn’t miss number 500. We’ll all look forward to another special, at 1000.

Liked by 1 person

• ivasallay said:

Life can keep us very busy at times, and that’s a good thing. Thank you for taking the time to read and comment!

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