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

2 is a digit in each one of those numbers. OEIS.org 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.

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.

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:

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?