## A Multiplication Based Logic Puzzle

### 792 Number Facts and Factors of the Year 2017

Happy New Year, everybody!

So much of what I’ve included in this post is a bit difficult for elementary school students, but here is an area problem that they should be able to do:

Perhaps you’ll recognize that as just another way to illustrate that 44² + 9² = 2017.

This area problem based on 33² + 28² + 12² = 2017 will be a little bit more challenging for students:

2016 had more factors, positive and negative, than anybody could have imagined, but 2017 is a prime number year, so hopefully it will be filled with less drama.

I have a lot to say about the number 2017 with a little help from twitter.

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2017 has only TWO positive real factors, 1 and 2017, and only TWO negative real factors, -1 and -2017. Positive or negative, ALL the real possible factor pairs for 2017 are

• 1 x 2017 and (-1) x (-2017).

2017 = 4(504) + 1. So 2017 is the sum of two squares. Which ones?

44² + 9² = 2017. That sum-of-squares number fact means that 2017 is the hypotenuse of a Pythagorean triple, specifically, 792-1855-2017. Since 2017 is a prime number, this triple is also a primitive.

Here’s how those numbers were calculated from the fact that  44² + 9² = 2017:

• 2(44)(9) = 792
• 44² – 9² = 1855
• 44² + 9² = 2017

2017 is also the short leg in a rather monstrous primitive Pythagorean triple:

• 2017² + 2,034,144² = 2,034,145²

Hmm. 44² + 9² = 2017 means we can find some COMPLEX factor pairs for 2017:

• (44 + 9i)(44 – 9i) = 2017
• (-44 + 9i)(-44 – 9i) = 2017
• (9 + 44i)(9 – 44i) = 2017
• (-9 + 44i)(-9 – 44i) = 2017

Who knows what all those COMPLEX factors will bring to the coming year? Each of them was derived from the fact that 44² + 9² = 2017.

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2017 is the sum of THREE squares three different ways:

• 37² + 18² + 18²  = 2017
• 33² + 28² + 12² = 2017
• 30² + 26² + 21² = 2017

Since 37² = 35² + 12², 30² = 24² + 18², and 26² = 24² + 10², we can write 2017 as the sum of these squares, too:

• 35² + 18² + 18² + 12² = 2017
• 30² + 24² + 21² + 10² = 2017
• 26² + 24² + 21² + 18² = 2017
• 24² + 24² + 21² + 18² + 10² = 2017

Here’s more sums and/or differences of squares from twitter:

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This representation of 2017 is “two” powerful:

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2017 is a PALINDROME in bases 31, 32, and 36:

• 232 BASE 31; note that 2(31²) + 3(31) + 2(1) = 2017
• 1V1 BASE 32 (V is 31 base 10); note that 1(32²) + 31(32) + 1(1) = 2017
• 1K1 BASE 36 (K is 20 base 10); note that 1(36²) + 20(36) + 1(1) = 2017

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Also Stetson.edu tells us this 2017 fact about the totient function: “2017 is a value of n for which φ(n) = φ(n-1) + φ(n-2).”

PROOF that 2017 is a prime number:

We can use these three facts to help us verify that 2017 is a prime number:

1. 2017 = 4(504) + 1
2. 44² + 9² = 2017
3. √2017 ≈ 44.8998886

Since 2017 can be written as one more than a multiple of four, and 44 and 9 have no common prime factors, 2017 will be a prime number unless it is divisible by 5, 13, 17, 29, 37, or 41 (all the prime numbers less than √2017 ≈ 44.9 that have a remainder of one when divided by 4). That’s right, we ONLY have to divide 2017 by those SIX numbers to verify that it is prime: (Read here for why this is true.)

• 2017 ÷ 5 = 403.4
• 2017 ÷ 13 ≈ 155.15
• 2017 ÷ 17 ≈ 118.65
• 2017 ÷ 29 ≈ 69.55
• 2017 ÷ 37 ≈ 54.51
• 2017 ÷ 41 ≈ 49.20

We don’t get a whole number answer for any of those divisions, so 2017 is prime!

If you’re looking for more reasons to be interested in the number 2017, read David Radcliffe’s article:

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Here are some other lovely mathematical thoughts about 2017 that I found on twitter:

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In case you don’t know (and apparently a lot of people don’t know), 0! = 1, so this expression really does equal 2017.

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And finally, if you click on this next link, David Mitchell will explain the tessellation of the number 2017.

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Since this is my 792nd post, I’ll write a little bit about the number 792:

792 has 24 factors so I’ll include a few of its many possible factor trees:

Since 792 has so many factors, it is a leg in MANY Pythagorean triples. Here I list some factors that generate PRIMITIVE triples:

• 2(396)(1): 792-156815-156817
• 2(99)(4): 792-9785-9817
• 2(44)(9): 792-1855-2017 (illustrated near the top of this post)
• 2(36)(11): 792-1175-1417

792 is the sum of three squares four different ways including three ways that repeat squares:

• 28² + 2² + 2² = 792
• 26² + 10² + 4² = 792
• 20² + 14² + 14² = 792
• 18² + 18² + 12² = 792

792 is a palindrome in bases 32 and 35:

• OO BASE 32 (O is 24 base 10); note that 24(32) + 24(1) = 792
• MM BASE 35 (M is 22 base 10); note that 22(35) + 22(1) = 792

Finally here is the factoring information for the number 792:

• 792 is a composite number.
• Prime factorization: 792 = 2 x 2 x 2 x 3 x 3 x 11, which can be written 792 = (2^3) x (3^2) x 11
• The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 x 3 x 2 = 24. Therefore 792 has exactly 24 factors.
• Factors of 792: 1, 2, 3, 4, 6, 8, 9, 11, 12, 18, 22, 24, 33, 36, 44, 66, 72, 88, 99, 132, 198, 264, 396, 792
• Factor pairs: 792 = 1 x 792, 2 x 396, 3 x 264, 4 x 198, 6 x 132, 8 x 99, 9 x 88, 11 x 72, 12 x 66, 18 x 44, 22 x 36 or 24 x 33

Taking the factor pair with the largest square number factor, we get √792 = (√36)(√22) = 6√22 ≈ 28.14249.

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