A Multiplication Based Logic Puzzle

Posts tagged ‘horse race’

800 Which Pony Will Take Second Place?

Every 100 posts I summarize the amount of factors of the previous 100 numbers.

MANY of the numbers from 701 to 800 have FOUR factors, and any other number-of-factors doesn’t even come close. For this Horse Race, SECOND place is much more interesting as there are several lead changes. I’ve shorten the track so the second place number-of-factors can reach the finish line.

So go ahead, pick the number-of-factors pony you think will come in SECOND place. Your best bets are 2, 6, 8, 12, 16 OR the second row of 4 factors!

Make your selection, then click on the graphic below to see how your pony does!

800-horse-race-01

Now let me tell you a little bit about the number 800.

800-prime-factorization

  • 800 is a composite number.
  • Prime factorization: 800 = 2 x 2 x 2 x 2 x 2 x 5 x 5, which can be written 800 = (2^5) x (5^2)
  • The exponents in the prime factorization are 5 and 2. Adding one to each and multiplying we get (5 + 1)(2 + 1) = 6 x 3 = 18. Therefore 800 has exactly 18 factors.
  • Factors of 800: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800
  • Factor pairs: 800 = 1 x 800, 2 x 400, 4 x 200, 5 x 160, 8 x 100, 10 x 80, 16 x 50, 20 x 40 or 25 x 32
  • Taking the factor pair with the largest square number factor, we get √800 = (√400)(√2) = 20√2 ≈ 28.28427.

800-factor-pairs

800 is the sum of four consecutive primes:

  • 193 + 197 + 199 + 211 = 800

800 is a palindrome in three different bases.

  • 2222 BASE 7 because 2(7^3) + 2(49) + 2(7) + 2(1) = 800 (This fact also means that x^3 + x² + x – 399 can be divided evenly by x – 7.)
  • 242 BASE 19 because 2(19²) + 4(19) + 2(1) = 800
  • PP BASE 31 (P is 25 base 10) because 25(31) + 25(1) = 800

800 is the sum of two squares two different ways:

  • 28² + 4² = 800
  • 20² + 20² = 800

That being true, it follows that 800 is the hypotenuse of two Pythagorean triples:

  • 480-640-800 which is 160 times 3-4-5
  • 224-768-800 which is 32 times 7-24-25

 

800 is also the sum of three squares:

  • 20² + 16² + 12² = 800

This chart summarizes the number of factors for the first 800 numbers and indicates that 39% of those numbers have square roots that can be simplified (reduced).

800-totals

In case you didn’t click on the Horse Race image before, here it is, no clicking required:

800 Horse Race

make science GIFs like this at MakeaGif

 

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700 Pick Your Pony! Who will win this Amount of Factors Horse Race?

  • 700 is a composite number.
  • Prime factorization: 700 = 2 x 2 x 5 x 5 x 7, which can be written 700 = (2^2) x (5^2) x 7
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 700 has exactly 18 factors.
  • Factors of 700: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700
  • Factor pairs: 700 = 1 x 700, 2 x 350, 4 x 175, 5 x 140, 7 x 100, 10 x 70, 14 x 50, 20 x 35 or 25 x 28
  • Taking the factor pair with the largest square number factor, we get √700 = (√100)(√7) = 10√7 ≈ 26.457513.

Because this is my 700th post, I think I’ll have another horse race. Some numbers from 601 to 700 have exactly 2 factors, 4 factors, and so forth up to 24 factors. (Only perfect squares can have an odd number of factors.)

Which number from 1 to 24 will win this amount of factors horse race? Which number will come in second place, or third place? Cheering for more than one pony will make the race even more interesting.

Here we see that the numbers 2, 6, & 8 are the first ones out of the gate. Click on the graphic to see the rest of this very thrilling horse race:

605

Every hundred posts I also like to focus on the percentage of numbers whose square roots can be simplified.

700 is divisible by 100 so its square root can easily be simplified: √700 = 10√7.

273 of the first 700 numbers have reducible square roots. That’s exactly 39%.

The rest of the numbers, 427, which is 41% of the first 700 numbers, do not have reducible square roots.

Here’s a table breaking down the amount of factors in each group of one hundred integers and the number of reducible square roots.

1-700 Amount of Factors

Here are some facts about the number 700.

700 is a palindrome in several bases:

  • 4A4 BASE 12; note A is equivalent to 1o in base 10, and 4(144) + 10(12) + 4(1) = 700
  • PP BASE 27; note P is equivalent to 25 in base 10, and 25(27) + 25(1) = 700
  • KK Base 34; note K is equivalent to 20 in base 10, and 20(34) + 20(1) = 700

700 is the sum of four consecutive prime numbers: 167 + 173 + 179 + 181.

Here is a beautiful painting of a horse race that I saw on twitter:

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