A Multiplication Based Logic Puzzle

The first six multiples of 642 are 642, 1284, 1926, 2568, 3210, and 3852.

2 is a digit in each one of those numbers. Stetson.edu reports that 642 is the smallest number that can make that claim.

  • 642 is a composite number.
  • Prime factorization: 642 = 2 x 3 x 107
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 642 has exactly 8 factors.
  • Factors of 642: 1, 2, 3, 6, 107, 214, 321, 642
  • Factor pairs: 642 = 1 x 642, 2 x 321, 3 x 214, or 6 x 107
  • 642 has no square factors that allow its square root to be simplified. √642 ≈ 25.3377.

642 is made from 3 different even numbers. I thought it might be fun to make a Venn diagram comparing 642 with other numbers made with the same three digits. I had never made a Venn diagram on a computer before so I first tried making one in Microsoft Word, but apparently the version of Word we have doesn’t allow any writing in the parts of the Venn diagram that intersect.

I looked online for a Venn diagram maker, but didn’t use any of them for various reasons.

Finally I made a Venn diagram using different colored circles in Paint to surround information I had copied from Excel. I had to redo the work in Excel and Paint several times, but it became easier and better looking with each attempt.

Permutations of 642 and Their Factors

I attempted to show in the Venn diagram that all six numbers are divisible by 2, 3, and 6, but I’m not sure that is clear looking at the diagram. I wondered if I was even making the Venn diagram correctly in every way. Having three circles can certainly complicate the diagram. I consulted a post on Purple Math on how to solve problems using Venn diagrams , but I’m still not 100% sure I made it correctly.

I looked at Wikipedia. It showed many different types of Venn diagrams including one that sorts letters of the Greek, Latin, and Cyrillic alphabets, but the diagram wasn’t labeled.

I also saw a great Venn diagram in a post for job seekers, but it contained no data.

Being confused, what could I do? I made a completely different Venn diagram this time using Microsoft Word.

Every counting number 3 or greater is part of at least one Pythagorean triple. A number being the hypotenuse doesn’t happen as often. An even number can only be the hypotenuse if at least one of its prime factors is also an hypotenuse.

Permutations of 642-no outline

13 is a hypotenuse, and its multiple, 624, is the hypotenuse of the Pythagorean triple 240-576-624.

41 is also a hypotenuse, and its multiple, 246, is the hypotenuse of the triple 54-240-246.

————————————————————–

An even number being part of a primitive Pythagorean triple also only happens half the time because only numbers divisible by 4 can be part of a primitive triple.

Of the six permutations of 6-4-2, only 264 and 624 are divisible by 4, so they are the only two that are part of any primitive triples. Each of them is part of four different primitive Pythagorean triples:

  • 264-1073-1105 calculated from 2(33)(4), 33² – 4², 33² + 4²
  • 264-1927-1945 calculated from 2(44)(3), 44² – 3², 44² + 3²
  • 264-17423-17425 calculated from 2(132)(1), 132² – 1², 132² + 1²
  • 23-264-265 calculated from 12² – 11², 2(12)(11), 12² + 11²
  • 624-1457-1585 calculated from 2(39)(8), 39² – 8², 39² + 8²
  • 624-10807-10825 calculated from 2(104)(3), 104² – 3², 104² + 3²
  • 624-97343-97345 calculated from 2(312)(1), 312² – 1², 312² + 1²
  • 407-624-745 calculated from 24² – 13², 2(24)(13), 24² + 13²

I wanted the circles on the Venn diagram to be outlined so I did only one edit to them. They looked amazing in Word, but when I cut and pasted them into Paint, this is how my picture looked:

Permutations of 642

It looks like making a Venn diagram with only two circles isn’t too difficult, but adding even one more circle makes it much more complicated. Typing anything in the intersecting areas also presents a challenge no matter how many circles are used, at least in the version of Word I used.

What experiences have you had making Venn diagrams?

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Comments on: "642 Was This Venn Diagram Made Correctly?" (4)

  1. I made this one in Word.

    Liked by 1 person

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