A Multiplication Based Logic Puzzle

672 can make MANY factor trees.  Here I’ve pictured only a few of the possibilities, one for each of its factor pairs (excluding 1 x 672).

Is it too soon to pick out a tree?

672 Factor Trees

Every one of those trees has the prime factors of 672: 2, 2, 2, 2, 2, 3, and 7, but finding them on each tree might be a challenge because I didn’t distinguish the prime factors from the other factors. Some of those prime factors might seem like they are lost in a pile of leaves. Can you find them on each tree?

672 is the 9th number to have 24 factors. Here is a number line highlighting all nine of those numbers and the distances between them.

24 Factors Number Line

Notice the difference between 672 and the previous number with 24 factors is 12, a record low.

You might get the impression looking at the number line that numbers having 24 factors might be much more common from now on. That may be true, nevertheless, the next number after 672 to have exactly 24 factors is 756 which is 84 numbers away and well past 720 the smallest number to have 30 factors.

Indeed infinitely many numbers have 24 factors, but probably 672 is the last one that will get much attention.

The numbers in the factor pair 24 and 28 are each exactly two numbers away from 26, their average. That means we are just 2² numbers away from 26².

In other words, 672 equals 26² – 2² which can be factored into (26 + 2)(26 – 2) so (26 + 2)(26 – 2) = 28 x 24 = 672.

  • 672 is a composite number.
  • Prime factorization: 672 = 2 x 2 x 2 x 2 x 2 x 3 x 7, which can be written 672 = (2^5) x 3 x 7
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 x 2 x 2 = 24. Therefore 672 has exactly 24 factors.
  • Factors of 672: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672
  • Factor pairs: 672 = 1 x 672, 2 x 336, 3 x 224, 4 x 168, 6 x 112, 7 x 96, 8 x 84, 12 x 56, 14 x 48, 16 x 42, 21 x 32, or 24 x 28
  • Taking the factor pair with the largest square number factor, we get √672 = (√16)(√42) = 4√42 ≈ 25.92296.

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