Because it is fall, and 640 has many factors, I decided to make factor trees using fall colors. Get out your rakes!

There are many other possible factor trees for 640, but raking leaves can be a lot of work, so I only made two of them.

640 is the hypotenuse of the Pythagorean triple 384-512-640.

OEIS.org informs us that 640 = 16!!!!!!, but if you type 16!!!!!! into a calculator, you will get an error message as soon as you type !!.

16!!!!!! ≠ (((((16!)!)!)!)!)!

There are 6 !’s so 16!!!!!! = 16(16-6)(16-12) = 16 x 10 x 4.

Here is this week’s Level 5 puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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- 640 is a composite number.
- Prime factorization: 640 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 640 = (2^7) x 5
- The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 640 has exactly 16 factors.
- Factors of 640: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640
- Factor pairs: 640 = 1 x 640, 2 x 320, 4 x 160, 5 x 128, 8 x 80, 10 x 64, 16 x 40, or 20 x 32
- Taking the factor pair with the largest square number factor, we get √640 = (√64)(√10) = 8√10 ≈ 25.298221

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OMG..I miss a day and there you go raking leaves and introducing MULTIFACTORIALS which I have never heard of, seen or suspected the existence of!!!!!! oops, didn’t mean to use so many exclamation signs…you are a bad influence.

I was about to ask how the 16!!!!!! got to be so really quite finite.

I hadn’t heard of it before, Paula hadn’t heard of it, and now you’re telling me you hadn’t either. It must not be all that well-known.

A few weeks ago I mentioned that 597!!! + 1 is a prime number.

597 is a large enough number and the number of factorials is small enough that 597!!! + 1 is way too big a number for any calculator I own to handle.

On the other hand 16 is a relatively small number and the number of factorials is big enough that 16!!!!!! only produces a 3-digit number.

I think it is the best way for a person to be a bad influence!!!!!