A Multiplication Based Logic Puzzle

Posts tagged ‘Factors’

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:

2017-find-the-area

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:

2017-area

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.

2017-triple

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:

792-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.

792-factor-pairs

791 and Level 4

To see if 791 is divisible by 7, you could try either one of these divisibility tricks:

  • 791 is divisible by 7 because 79 – 2(1) = 77 which obviously is divisible by 7.
  • 791 is divisible by 7 because 79 + 5(1) = 84 which most people know is 12 × 7.

791-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 791 is a composite number.
  • Prime factorization: 791 = 7 x 113
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 791 has exactly 4 factors.
  • Factors of 791: 1, 7, 113, 791
  • Factor pairs: 791 = 1 x 791 or 7 x 113
  • 791 has no square factors that allow its square root to be simplified. √791 ≈ 28.12472222.

791-factor-pairs

790 and Level 3

790’s prime factorization is 2 × 5 × 79. Since two of those numbers, 5 and 79, have a remainder of one when divided by four, I automatically know that 790 is the hypotenuse of exactly four Pythagorean triples.

790-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 790 is a composite number.
  • Prime factorization: 790 = 2 x 5 x 79
  • 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 790 has exactly 8 factors.
  • Factors of 790: 1, 2, 5, 10, 79, 158, 395, 790
  • Factor pairs: 790 = 1 x 790, 2 x 395, 5 x 158, or 10 x 79
  • 790 has no square factors that allow its square root to be simplified. √790 ≈ 28.106939.

790-factor-pairs

789 and Level 2

789 consists of exactly three consecutive numbers so it is divisible by 3.

789-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 789 is a composite number.
  • Prime factorization: 789 = 3 x 263
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 789 has exactly 4 factors.
  • Factors of 789: 1, 3, 263, 789
  • Factor pairs: 789 = 1 x 789 or 3 x 263
  • 789 has no square factors that allow its square root to be simplified. √789 ≈ 28.08914.

789-factor-pairs

788 and Level 1

Since 88, its last two digits, are divisible by 4, we know that 788 and every other whole number ending in 88 is divisible by 4.

I learned the following fascinating fact about these six numbers starting with 788 from Stetson.edu:

788-consecutive-numbers

788 is also palindrome 404 in BASE 14. Note that 4(196) + 0(14) + 4(1) = 788.

788-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

  • 788 is a composite number.
  • Prime factorization: 788 = 2 x 2 x 197, which can be written 788 = (2^2) x 197
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 788 has exactly 6 factors.
  • Factors of 788: 1, 2, 4, 197, 394, 788
  • Factor pairs: 788 = 1 x 788, 2 x 394, or 4 x 197
  • Taking the factor pair with the largest square number factor, we get √788 = (√4)(√197) = 2√197 ≈ 28.071338.

788-factor-pairs

786 and Level 5

786 is even so it is divisible by 2. Also since 786 is made from 3 consecutive numbers, we can tell automatically that it is divisible by 3. Those two facts together mean 786 is also divisible by 6.

  • 786 is a composite number.
  • Prime factorization: 786 = 2 x 3 x 131
  • 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 786 has exactly 8 factors.
  • Factors of 786: 1, 2, 3, 6, 131, 262, 393, 786
  • Factor pairs: 786 = 1 x 786, 2 x 393, 3 x 262, or 6 x 131
  • 786 has no square factors that allow its square root to be simplified. √786 ≈ 28.03569.

786-factor-pairs

Today’s Find the Factors puzzle:

786-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s a little more about the number 786:

786 is 123 in BASE 27 because 1(27²) + 2(27) + 3(1) = 786.

786 is the sum of two consecutive primes: 389 + 397 = 786

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786-logic

785 and Level 4

  • 785 is a composite number.
  • Prime factorization: 785 = 5 x 157
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 785 has exactly 4 factors.
  • Factors of 785: 1, 5, 157, 785
  • Factor pairs: 785 = 1 x 785 or 5 x 157
  • 785 has no square factors that allow its square root to be simplified. √785 ≈ 28.01785.

785-factor-pairs

Now for today’s puzzle:

785-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s a little more about the number 785:

785 is the sum of two squares two different ways:

  • 28² + 1² = 785
  • 23² + 16² = 785

785 is also the sum of three squares three different ways:

  • 26² + 10² + 3² = 785
  • 25² + 12² + 4² = 785
  • 19² + 18² + 10² = 785

Because its prime factorization is 5 × 157 (two numbers that are also the sum of two squares), 785 is the hypotenuse of four Pythagorean triples, two of which are primitive triples:

  • 56-783-785 primitive calculated from 2(28)(1), 28² – 1², 28² + 1²
  • 273-736-785 primitive calculated from 23² – 16², 2(23)(16), 23² + 16²
  • 425-660-785 which is 5 times 85-132-157
  • 471-628-785 which is 157 times 3-4-5

785 is also a palindrome in two different bases:

  • 555 BASE 12; note that 5(144) + 5(12) + 5(1) = 785
  • 101 BASE 28; note that 1(28²) + 0(28) + 1(1) = 785

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785-logic

 

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