A Multiplication Based Logic Puzzle

Archive for the ‘Mathematics’ Category

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

780 is the 39th Triangular Number

  • 780 is a composite number.
  • Prime factorization: 780 = 2 x 2 x 3 x 5 x 13, which can be written 780 = (2^2) x 3 x 5 x 13
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 780 has exactly 24 factors.
  • Factors of 780: 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780
  • Factor pairs: 780 = 1 x 780, 2 x 390, 3 x 260, 4 x 195, 5 x 156, 6 x 130, 10 x 78, 12 x 65, 13 x 60, 15 x 52, 20 x 39, or 26 x 30
  • Taking the factor pair with the largest square number factor, we get √780 = (√4)(√195) = 2√195 ≈ 27.92848.

780-factor-pairs

There are MANY ways to make factor trees for 780. Here are just three of them:

780 Factor Trees

Stetson.edu informs us that (7 + 5)(8 + 5)(0 + 5) = 780.

780 is the sum of consecutive prime numbers two different ways:

  • 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 780 (10 consecutive primes).
  • 191 + 193 + 197 + 199 = 780 (4 of my favorite consecutive primes).

26 and 30 are both exactly 2 numbers away from their average 28, so 780 can be written as the difference of two squares:

780 = 26 x 30 = (28 – 2)(28 + 2) = 28² – 2² = 784 – 4.

(It can be written as the difference of two squares three other ways, but I won’t list them here.)

Because 5 and 13 are both factors of 780, it is the hypotenuse of FOUR Pythagorean triangles making each of these equations true:

  • 192² + 756² = 780²
  • 300² + 720² = 780²
  • 396² + 672² = 780²
  • 468² + 624² = 780²

780 can be written as the sum of three perfect squares two different ways:

  • 26² + 10² + 2² = 780
  • 22² + 14² + 10² = 780

780 BASE 10 is palindrome QQ BASE 29. (Q is 26 in BASE 10.) Note that 26(29) + 26(1) = 780.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 780, making 780 the 39th triangular number. Since 20 is in the exact middle of that list of numbers, 780 is also the 20th hexagonal number.

Triangular numbers are interesting, but are they good for anything? Here’s one good thing:

Count the Terms of Sums Squared

There are 26 letters in the English alphabet. 39 variables would use the alphabet exactly one and one half times, but that’s okay because we can use upper case letters the second time around:

(a + b + c + . . . + x + y+ z + A + B + C + . . . + K + L + M)² has exactly 780 terms because it has 39 single variables, and 780 is the 39th triangular number.

780 is the 39th triangular number because 39⋅40/2 = 780.

A couple of months ago I saw a fascinating image on twitter. It is a square made with 1001 dots representing the numbers from -500 to +500 with zero in the exact middle. The triangular numbers are represented by the “+” pattern seen here. If we added about 600 more dots to the square, -780 and +780 would also lie on that +.

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Half of all triangular numbers are also hexagonal numbers. If there were enough dots, then -780 and +780 would also be among the dots forming the “-” pattern representing the hexagonal numbers. 780 is the 20th hexagonal number because 20(2⋅20-1) = 780.

 

How Lucky Can 777 Be?

  • 777 is a composite number.
  • Prime factorization: 777 = 3 x 7 x 37
  • 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 777 has exactly 8 factors.
  • Factors of 777: 1, 3, 7, 21, 37, 111, 259, 777
  • Factor pairs: 777 = 1 x 777, 3 x 259, 7 x 111, or 21 x 37
  • 777 has no square factors that allow its square root to be simplified. 77727.8747197.

777-factor-pairs

Some people think that 7 is a lucky number. If that is true, then 777 should be even luckier.

Some numbers are lucky enough to be included in Multiplication Rhymes.ppt – mathval, a fun power point that helps students learn 12 multiplication facts, including these three that use Lucky Numbers:

  • 3 & 7 are always lucky numbers; 3 x 3 = 9 lives of a cat.
  • 3 & 7 are always lucky numbers; 3 x 7 = 21 lucky age.
  • 3 & 7 are always lucky numbers; 7 x 7 = 49er Gold Miner.

In Number Theory Lucky Numbers are actually defined and can be generated using a sieve somewhat similar to the prime number generating Sieve of Eratosthenes. There is an infinite number of Lucky Numbers, and yes, 3, 7, 9, 21, 49, and 777 all make the list.

Oeis.org’s wiki, Lucky numbers, includes several lucky number lists including the first 33 composite Lucky Numbers thus defined because ALL of their factors are also Lucky Numbers. 777 was the 19th number on that particular list because ALL of its factors, 1, 3, 7, 21, 37, 111, 259, 777 are Lucky Numbers, too!

If that isn’t lucky enough, 777 is a repdigit in three different bases.

  • 3333 BASE 6; note that 3(6^3) + 3(6^2) + 3(6^1) + 3(6^0) = 777
  • 777 BASE 10; note that 7(100) + 7(10) + 7(1) = 777
  • LL BASE 36 (L is 21 base 10); note that 21(36) + 21(1) = 777

Did you notice that lucky numbers 3, 7, and 21 showed  up again? I liked that coincidence so much that I made this graphic:

777 Repdigit

777 is also the sum of three squares four different ways:

  • 26² + 10² + 1² = 777
  • 22² + 17² + 2² = 777
  • 20² + 19² + 4² = 777
  • 20² + 16² + 11² = 777

 

729 What I Did for Paula Krieg’s Star Project

The 12-Fold Rosette has its origin in Islamic Geometry. Paula Krieg is asking that people from a variety of backgrounds color in a section of the rosette and send it back to her so that she can assemble the different pieces into one lovely unified rosette. We are all stars, and some of the pieces look like stars, so she calls it her star project. You are invited to become a part of the project as well. Just click on this invitation to learn more. You can get a tile by either leaving a comment or e-mailing her a request, but do it soon. You will need to return your tile to her by 10 January 2016 to be included in the project.

A partially filled 12-fold Rosette

Rosette from Paula Krieg’s blog showing some possible coloring schemes for the tiles.

The piece that Paula sent me looked like this:

Rosette tile 21

I thought I would color it black, gray, white, and red because I think they look good together.

Since I don’t have any crayons in my house right now, and all my coloring pencils need to be sharpened, I decided to color it on my computer using the program Paint.

I brought up the e-mail Paula, opened the attachment, hit “Print Screen”, copied and pasted it into Paint, then cropped it.

I soon discovered that I could color in my entire tile using only black and gray and stay true to a goal I had given myself. The very tiny triangles insisted on staying white, but if I had increased the view size before I hit “Print Screen”, I would have been able to color them, too.

Tile 21

I saved my coloring, but immediately felt like it really should have some red in it. I brought up my saved work, clicked on “save as”, named it “red tile”, and added some red. However, saving the file changed how it now received color. When I added some red, it looked like this, and I liked it and saved it again:

Tile 21 red

Then I thought I might want a little more white, so I brought it back up and did “save as” again before adding some. This time it seem to allow even less additional color in each shape. (In other words, we see much more gray than black.)

Tile 21 red and white

 

I had so much fun doing this project. I have never considered myself to be much of an artist, but I felt a little artistic doing this project. I like all three tiles, but I think I like the final product the best.

You should join the project, too! You’re a star! Go ahead and shine! Whatever colors you use or whatever method you use to color your tile, your efforts will be appreciated. Also, because everyone is unique, every tile will look at least a little different. No tile will be ordinary.

Likewise, you might think that 729 is just an ordinary number, but much depends on how you look at it, too. Here are 729 equally sized squares.

729 Equal Squares

There’s a variety of ways to count the squares. Some ways remind us that 729 is a perfect square, other ways reminds us 729 is a perfect cube, and perhaps you can even find a way to see that 729 is a perfect 6th power, too.

279 is a perfect square, cube, and 6th power

729 can be written as the sum of consecutive numbers several different ways:

  • 364 + 365 = 729; that’s 2 consecutive numbers.
  • 242 + 243 + 244 = 729; that’s 3 consecutive numbers.
  • 119 + 120 + 121 + 122 + 123 + 124 = 729; that’s 6 consecutive numbers.
  • 77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85 = 729; that’s 9 consecutive numbers.
  • 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 = 729; that’s 18 consecutive numbers.
  • 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 = 729; that’s 27 consecutive numbers.

729 is 1,000,000 in base 3, and 1000 in base 9, and 100 in base 27.

729 is also a palindrome in two bases:

  • 1331 BASE 8; note that 1(512) + 3(64) + 3(8) + 1(1) = 729.
  • 121 BASE 26; note that 1(26^2) + 2(26) + 1(1) = 729.

It is interesting that one less than 9 is 8, and one less than 27 is 26, and those are the two bases in which 729 is a palindrome!

Here is the factoring information for the number 729:

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  • 729 is a composite number. 729 = 27^2; 729 = 9^3; and 729 = 3^6
  • Prime factorization: 729 = 3 x 3 x 3 x 3 x 3 x 3, which can be written 729 = (3^6)
  • The exponent in the prime factorization is 6. Adding one we get (6 + 1) = 7. Therefore 729 has exactly 7 factors.
  • Factors of 729: 1, 3, 9, 27, 81, 243, 729
  • Factor pairs: 729 = 1 x 729, 3 x 243, 9 x 81, or 27 x 27
  • 729 is a perfect square √729 = 27.

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728 Number Facts and Factors of the Year 2016

The year 2016 will have some fun mathematical properties:

2016 is the sum of all the counting numbers from 1 to 63. That makes 2016 the 63rd triangular number which can be calculated rather quickly using 63 x 64/2 = 2016. That is definitely an example of multiplication being a shortcut for addition!

When was the last time a year was a triangular number? 1953. That’s the sum of the first 62 numbers, and it is also the year that I was born.

Besides being the sum of the counting numbers from 1 to 63, 2016 is the sum of consecutive numbers a few other ways as well:

  • 671 + 672 + 673 = 2016; that’s 3 consecutive numbers.
  • 285 + 286 + 287 + 288 + 289 + 290 + 291 = 2016; that’s 7 consecutive numbers.
  • 220 + 221 + 222 + 223 + 224 + 225 + 226 + 227 + 228 = 2016; that’s 9 consecutive numbers.
  • 86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103 + 104 + 105 + 106 = 2016; that’s 21 consecutive numbers.

2016 is the sum of the eighteen prime numbers from 71 to 157.

2016 sum of consecutive primes

2016 is also the 32nd hexagonal number because 2 x 32² – 32 = 2016. (All hexagonal numbers are also triangular numbers, and half of all hexagonal numbers are triangular numbers.)

From Stetson.edu we learn that the sum of the square and cube of 2016 is a number containing all the digits 0 – 9 exactly once:

sum of square and cube of 2016

2016 is the short leg in these four primitive Pythagorean triples:

  • 2016-3713-4225
  • 2016-12,463-12,625
  • 2016-20,687-20,785
  • 2016-1,016,063-1,016,065

2016 is not a palindrome in any base until base 47. We just need a symbol to represent 42 base 10 in base 47 because 42(47) + 42(1) = 2016.

Thank you Slate magazine  for including my post in an article about properties of 2016. Also thank you for referring me to eljjdx.canalblog.com which has some very interesting information about the number 2016 that I didn’t include in this post. Something on my computer did a great job translating both of those articles from French into English.

Reflexivemaths has also written many thoughtful starter questions that explore the number 2016.

Before I give the factors of 2016, Let me share a few fun number facts about the number 2016 that I saw on twitter:

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That is the most complicated use of combinations I have ever seen. The innermost parenthesis means 4!/(2!∙2!) which equals 6. Since 2 was raised to that power, we get 2^6 = 64. Then 64!/(62!∙2!) = 2016.

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The last two years on New Year’s Eve I have predicted the factors of the coming year. Each time my predictions were 100% accurate. I am now ready to make my predictions for the factors of 2016:

(Drum roll) There will be 36 positive factors for 2016:

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016. Sadly, if you multiply any of them by -1, you’ll know a negative factor for 2016, too.

The factors of 2016 will also come in pairs: 1 x 2016, 2 x 1008, 3 x 672, 4 x 504, 6 x 336, 7 x 288, 8 x 252, 9 x 224, 12 x 168, 14 x 144, 16 x 126, 18 x 112, 21 x 96, 24 x 84, 28 x 72, 32 x 63, 36 x 56and 42 x 48.

2016-factor-pairs

Because the number of 2016’s prime factors is a power of two, 2016 can make a nicely proportioned factor tree especially if we use any of the factor pairs that are in red. (14 x 114 is in bold only because I like the way it looks.) Because 2016 has 8 prime factors, we can get a very full and impressive tree. For example:

Tree 2016

Since this is my 728th post, I’ll write a little about the number 728, too. I’ll start with this factor tree for 728:

Tree 728

It’s not as impressive as the factor tree for 2016, but did you notice that 2016 and 728 share several of the same prime factors?

When two numbers share some of the same factors, we may wonder what is the greatest common factor and what is the least common multiple?

One way to find either value is to use the prime factorization of both numbers. The prime factorization of 2016 uses three bases: 2, 3, and 7, while the prime factorization for 728 uses these bases: 2, 7, and 13. Write down all those bases without duplication, and you get 2, 3, 7, and 13.

For the GREATEST COMMON FACTOR (GCF), look at the prime factorizations and choose the SMALLEST exponent that appears in each. The greatest common factor cannot be bigger than the smallest number, 728. In this case it is 56 which will divide evenly into both 728 and 2016.

GCF and LCM 728, 2016

For the LEAST COMMON MULTIPLE (LCM), look at the prime factorizations and choose the LARGEST exponent that appears in each. The least common multiple cannot be smaller than the largest number, 2016. In this case it is 26,208 which both 728 and 2016 can divided into evenly.

This method for finding the GCF and LCM will also work for three, four, or more numbers and even variable bases like x, y, or z. I encourage you to give it a try!

Here are some more number facts about the number 728:

26 x 28 = 728, and both 26 and 28 are one number away from 27, their average, so 728 is one number away from 27².

Here’s proof: 26 x 28 = (27 – 1)(27 + 1) = (27² – 1²) = 729 – 1 = 728.

Because 13 is one of its prime factors, 728 is the hypotenuse of Pythagorean triple 280-672-728. The greatest common factor of those three numbers is the same as the greatest common factor of 728 and 2016, but what is their least common multiple? It turns out to be 5 x 12 x 13 x 56, the product of the numbers in the primitive Pythagorean triple, 5-12-13, and 56. Using prime factorizations we get (2^5)(3^1)(5^1)(7^1)(13^1). We get the same answer using either method.

728 is a palindrome (repdigit) in three other number bases:

  • 728 is 222222 BASE 3; note that 2(3^5) + 2(3^4) + 2(3^3) + 2(3^2) + 2(3^1) + 2(3^0) = 728.
  • 888 BASE 9; note that 8(81) + 8(9) + 8(1) = 728.
  • QQ BASE 27 (Q = 26 base 10); note that 26(27) + 26(1) = 728.

Here is more factoring information for 728:

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  • 728 is a composite number.
  • Prime factorization: 728 = 2 x 2 x 2 x 7 x 13, which can be written 728 = (2^3) x 7 x 13
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 728 has exactly 16 factors.
  • Factors of 728: 1, 2, 4, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, 728
  • Factor pairs: 728 = 1 x 728, 2 x 364, 4 x 182, 7 x 104, 8 x 91, 13 x 56, 14 x 52, or 26 x 28
  • Taking the factor pair with the largest square number factor, we get √728 = (√4)(√182) = 2√182 ≈ 26.981475.

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720 Christmas Factor Trees

720 has more factors than any previous number. It has 30 factors.

6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 720, and is usually written 6! = 720.

Since I hosted a Christmas Eve dinner for my extended family, I thought of some of the many ways people could fill up their plates.

720 Dinners

If we made a tree diagram of all the possible dinners that could be made with exactly one item from each column, it would contain 720 lines and require quite a few pages.

The fundamental counting principle tells us the easiest way to count all those dinners is to multiply together the number of items in each column. In this case that would be 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 720.

On the other hand factor trees for 720 are easy to make.

Here are some of MANY possible factor trees for 720:

720 Some Factor Trees

None of those were very attractive, but here are some that are much better looking:

720 More Factor Trees

Here are some other random facts about the number 720:

Because 5 is one of its factors, 720 is the hypotenuse of the Pythagorean triple 432-576-720. What is the greatest common factor of those 3 numbers? The greatest common factor is in the factor pair with the number 5. All 15 factor pairs for 720 are listed at the end of the post.

720 Hypotenuse

The sum of the interior angles of any hexagon is 720 degrees.

720 degrees hexagons

720 is a palindrome in three different bases:

  • 5A5 in base 11; note that 5(121) + 10(11) + 5(1) = 720.
  • OO in base 29; (O is 24 base 10) note that 24(29) + 24(1) = 720.
  • KK in base 35; (K is 20 base 10) note that 20(35) + 20(1) = 720.

 

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  • 720 is a composite number.
  • Prime factorization: 720 = 2 x 2 x 2 x 2 x 3 x 3 x 5, which can be written 720 = (2^4) x (3^2) x 5
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each and multiplying we get (4 + 1)(2 + 1)(1 + 1) = 5 x 3 x 2 = 30. Therefore 720 has exactly 30 factors.
  • Factors of 720: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720
  • Factor pairs: 720 = 1 x 720, 2 x 360, 3 x 240, 4 x 180, 5 x 144, 6 x 120, 8 x 90, 9 x 80, 10 x 72, 12 x 60, 15 x 48, 16 x 45, 18 x 40, 20 x 36 or 24 x 30

Taking the factor pair with the largest square number factor, we get √720 = (√144)(√5) = 12√5 ≈ 26.8328157.

 

714 Home Runs

Babe Ruth hit 714 home runs. His record for the most home runs stood for nearly 39 years when Hank Aaron hit the 715th home run of his career.

The number of prime factors that 714 has is a power of two so 714 can make a very attractive factor tree all by itself. 715 only has three prime factors so its factor tree doesn’t look as nice. I still wanted to make factor trees for both of them so I multiplied them together and made this impressive looking factor tree:

714 and 715 Baseballs and Factor Trees

(The picture of the 714 baseball in the graphic above listed e-bay as the source while the picture of the 715 baseball was from Sports Memorabilia.com. The 715 ball is autographed by Al Downing, the former all-star and comeback player of the year, who pitched the ball that became Hank Aaron’s 715th home run.)

The numbers 714 and 715 form what is now called a Ruth Aaron Pair. Other consecutive numbers whose factors add up to equal numbers are also given that name.

Between the numbers 714 and 715 ALL the prime numbers less than or equal to 17 are used as factors exactly one time.

I became aware of these facts several months ago when Dan Bach sent me this tweet:

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This is my 714th post so I will focus on the number 714 and some different ways it hits a home run:

714 can be written as the sum of consecutive numbers many ways:

  • 237 + 238 +239 = 714; that’s 3 consecutive numbers.
  • 177 + 178 + 179 + 180 = 714; that’s 4 consecutive numbers.
  • 99 + 100 + 101 + 102 + 103 + 104 + 105 = 714; that’s 7 consecutive numbers.
  • 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 = 714; that’s 12 consecutive numbers.
  • 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 = 714; that’s 17 consecutive numbers.
  • 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 = 714; that’s 21 consecutive numbers.

714 is the sum of the twelve prime numbers from 37 to 83.

714 is also the sum of the fourteen prime numbers from 23 to 79. Can you name all those prime numbers?

Because 17 is one of its factors, 714 is the hypotenuse of the Pythagorean triple 336-630-714. What is the greatest common factor of those three numbers?

Stetson.edu alerted me to the fact that 714 is the smallest number that uses all possible digits exactly the same number of times in both BASE 2 and BASE 5:

  • 714 is 1011001010 in BASE 2; that’s five 1’s and five 0’s, the only digits used in BASE 2.
  • 714 is 10324 in BASE 5; note that 0, 1, 2, 3, and 4 are the only digits used in BASE 5, and each is used exactly once.

In addition 714 is a palindrome in two bases:

  • 181 BASE 23; note that 1(23²) + 8(23) + 1(1) = 714.
  • LL BASE 33 (L is 21 in base 10); note that 21(33) + 21(1) = 714.

Here is the factoring information for the number 714:

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  • 714 is a composite number.
  • Prime factorization: 714 = 2 x 3 x 7 x 17
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 714 has exactly 16 factors.
  • Factors of 714: 1, 2, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714
  • Factor pairs: 714 = 1 x 714, 2 x 357, 3 x 238, 6 x 119, 7 x 102, 14 x 51, 17 x 42, or 21 x 34
  • 714 has no square factors that allow its square root to be simplified. √714 ≈ 26.720778.

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After I wrote this post, I tweeted Dan Bach about it, and he let me know how he first learned about Ruth Aaron pairs:

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