## A Multiplication Based Logic Puzzle

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

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As you read this post, you may get the impression that 2016 is a perfect number. All perfect numbers are triangular numbers AND hexagonal numbers, but 2016 is still NOT a perfect number. Here’s why:

The sum of the first 31 factors of 2016 is

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 = 2016

but if you add in its other four factors 336 + 504 + 672 + 1008, you get 4536 which is way too much to make 2016 a perfect number.

True, 2016 is (2^5)(2^6 – 1) = 32 x 63, and all perfect numbers can be written similarly with consecutive powers of 2, BUT the odd factor, 63, is not a PRIME number so 2016 is not a perfect number.

IF 63 WERE a prime number, then the sum of the factors of 2016 would be

1 + 2 + 4 + 8 + 16 + 32 + 63 + 126 + 252 + 504 + 1008 = 2016

But again, 63 is NOT a prime number so 3, 6, 7, 9, 12, 14, 18, 21, 24, 28, 36, 42, 48, 56, 72, 84, 96, 112, 144, 168, 224, 288, 336, and 672 are also factors of 2016, and 2016 is NOT a perfect number.

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Even though 2016 is NOT a perfect number, it is still an exciting number in many more ways:

2016 = (10 × 9 × 8 × 7 × 6)/(5 + 4 + 3 + 2 + 1)

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:

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 all the factors we can expect to see in the year 2016, let me share a few fun number facts that I saw on twitter:

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.

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:

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:

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.

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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### 727 Enjoy Some Sparkling Cider!

Have a happy New Year’s Eve! My husband and I will be enjoying some Sparkling Apple-Pear. I invite you to have some, too.

Perhaps the goblet in the puzzle can start off your festivities.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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• 727 is a prime number.
• Prime factorization: 727 is prime and cannot be factored.
• The exponent of prime number 727 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 727 has exactly 2 factors.
• Factors of 727: 1, 727
• Factor pairs: 727 = 1 x 727
• 727 has no square factors that allow its square root to be simplified. √727 ≈ 26.9629375.

727 is a palindrome in base 10 and one other base:

• 727 BASE 10; note that 7(100) + 2(10) + 7(1) = 727.
• 1B1 BASE 22 (B = 11 base 10); note that 1(22²) + 11(22) + 1(1) = 727.

Stetson.edu informs us that 727² = 528529, a number whose digits can be split in half to make two consecutive numbers.

Since 727 is a prime number, there is only one way it can be expressed as the sum of consecutive numbers: 363 + 364 = 727.

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### Simplifying √726

• 726 is a composite number.
• Prime factorization: 726 = 2 x 3 x 11 x 11, which can be written 726 = 2 x 3 x (11^2)
• The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 726 has exactly 12 factors.
• Factors of 726: 1, 2, 3, 6, 11, 22, 33, 66, 121, 242, 363, 726
• Factor pairs: 726 = 1 x 726, 2 x 363, 3 x 242, 6 x 121, 11 x 66, or 22 x 33
• Taking the factor pair with the largest square number factor, we get √726 = (√121)(√6) = 11√6 ≈ 26.944387.

Most numbers whose square roots can be simplified are divisible by 4, 9 or both. If I were trying to simplify √726, I would first apply divisibility tricks for 4 and 9 to the number 726:

• 726 is even so it is divisible by 2, but since 26 is not divisible by 4, neither is 726.
• Also notice that 7 + 2 + 6 = 15, a number divisible by 3, but not by 9. That means 726 is divisible by 3, but not by 9.

Thus 726 is NOT divisible by either 4 or 9. I will make a little cake to see if it is divisible by any other square numbers.

Dividing by 2 and then by 3 could be a good place to start, but rather than do those two divisions, I would simply do one division as I divide 726 by 6:

I would immediately recognize that 121 is 11 × 11, so I don’t need to make any more layers for this factoring cake!

To simplify √726, take the square root of everything on the outside of this cake (the numbers in red): Thus, √726 = (√6)(√121) = 11√6.

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This level 3 puzzle is not difficult to do. If you start at the top of the first column and work down cell by cell, you’ll catch the rhythm of finding all the factors that can make this puzzle function as a multiplication table:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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Here are a few more facts about the number 726:

726 is the average of 11^2 and 11^3 which is a characteristic that also makes it the 11th Pentagonal pyramidal number.

{724, 725, 726} is only the second set of three consecutive numbers in which each number in the set is the product of exactly one number squared and one different number. {603, 604, 605} was the first set.

726 is the sum of consecutive prime numbers 359 and 367.

726 is a palindrome in three bases:

• 10401 BASE 5; note that 1(625) + 0(125) + 4(25) + 0(5) + 1(1) = 726.
• 141 BASE 25; note that 1(625) + 4(25) + 1(1) = 726.
• MM BASE 32 (M = 22 base 10); note that 22(32) + 22(1) = 726.

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### 725 and Level 2

• 725 is a composite number.
• Prime factorization: 725 = 5 x 5 x 29, which can be written 725 = (5^2) x 29
• 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 725 has exactly 6 factors.
• Factors of 725: 1, 5, 25, 29, 145, 725
• Factor pairs: 725 = 1 x 725, 5 x 145, or 25 x 29
• Taking the factor pair with the largest square number factor, we get √725 = (√25)(√29) = 5√29 ≈ 26.925824.

Give this Level 2 puzzle a try!

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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Perhaps, you would like to know some other facts about the number 725:

725 can be expressed as the sum of consecutive numbers five different ways:

• 362 + 363 = 725; that’s 2 consecutive numbers.
• 143 + 144 + 145 + 146 + 147 = 725; that’s 5 consecutive numbers.
• 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 = 725; that’s 10 consecutive numbers.
• 17 + 18 + 19 + . . . + 29 + . . . + 39 + 40 + 41 = 725; that’s 25 consecutive numbers.
• 11 + 12 + 13 + . . . + 25 + . . . + 37 + 38 + 39 = 725; that’s 29 consecutive numbers.

725 is also the sum of the eleven prime numbers from 43 to 89.

The factors in one of its factor pairs, 25 x 29, are both 2 numbers away from their average, 27, so 725 is just 4 numbers away from perfect square 27² = 729 . Thus, 25 x 29 =  (27 – 2)(27 + 2) = 27² – 2² = 729 – 4 = 725.

725 is the sum of two squares three different ways:

• 26² + 7² = 725
• 25² + 10² = 725
• 23² + 14² = 725

Because ALL of its prime factors have a remainder of one when divided by four, 725 is the hypotenuse of primitive Pythagorean triples:

• 364-627-725 which was calculated using 2(26)(7), 26² – 7², 26² + 7²
• 333-644-725 which was calculated using 23² – 14², 2(23)(14), 23² + 14²

It is also the hypotenuse of FIVE other Pythagorean triples.

• 85-720-725
• 120-715-725
• 203-696-725
• 435-580-725
• 500-525-725

725 is a palindrome in two bases:

• 505 BASE 12; note that 5(144) + 0(12) + 5(1) = 725.
• PP BASE 28 (P = 25 base 10); note that 25(28) + 25(1) = 725.

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### 724 and Level 1

• 724 is a composite number.
• Prime factorization: 724 = 2 x 2 x 181, which can be written 724 = (2^2) x 181
• 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 724 has exactly 6 factors.
• Factors of 724: 1, 2, 4, 181, 362, 724
• Factor pairs: 724 = 1 x 724, 2 x 362, or 4 x 181
• Taking the factor pair with the largest square number factor, we get √724 = (√4)(√181) = 2√181 ≈ 26.907248.

Solving this Level 1 puzzle will be even easier:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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Here is a little more about the number 724:

87 + 88 + 89 + 90 + 91 + 92 + 93 + 94 = 724; that’s 8 consecutive numbers.

724 is the sum of the 6 prime numbers from 107 to 137.

724 is also the sum of the 4 primes from 173 to 191.

18² + 20² = 724.

Because 181 is one of its factors, 724 is the hypotenuse of the Pythagorean triple 76-720-724. What 1-digit number is the greatest common factor of those three numbers?

724 is palindrome 202 in BASE 19; note that 2(19²) + 0(19) + 2(1) = 724.

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### 723 and Level 6

7 + 2 + 3 = 12, a multiple of 3, so 723 is divisible by 3.

• 723 is a composite number.
• Prime factorization: 723 = 3 x 241
• 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 723 has exactly 4 factors.
• Factors of 723: 1, 3, 241, 723
• Factor pairs: 723 = 1 x 723 or 3 x 241
• 723 has no square factors that allow its square root to be simplified. √723 ≈ 26.88866.

Finding the factors to solve this Level 6 puzzle will be more challenging than that. . .

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here is a little more about the number 723:

723 is the sum of consecutive numbers three different ways:

• 361 + 362 = 723; that’s 2 consecutive numbers.
• 240 + 241 + 242 = 723; that’s 3 consecutive numbers.
• 118 + 119 + 120 + 121 + 122 + 123 = 723; that’s 6 consecutive numbers.

Because 241 is one of its factors, 723 is the hypotenuse of the Pythagorean triple 360-627-723. What is the greatest common factor of those three numbers? (It is one of the four factors of 723.)

723 is repdigit and palindrome 333 in BASE 15; note that 3(15²) + 3(15) + 3(1) = 723.

Stetson.edu informs us that  (1!)! + (2!)! + (3!)! = 723. That looks more exciting than it really is. It is only equivalent to saying 1 + 2 + 720 = 723.

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Thank you, Ricardo, for completing the puzzle and tweeting your work:

### 722 and Level 5

• 722 is a composite number.
• Prime factorization: 722 = 2 x 19 x 19, which can be written 722 = 2 x (19^2)
• The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 722 has exactly 6 factors.
• Factors of 722: 1, 2, 19, 38, 361, 722
• Factor pairs: 722 = 1 x 722, 2 x 361, or 19 x 38
• Taking the factor pair with the largest square number factor, we get √722 = (√361)(√2) = 19√2 ≈ 26.870058.

Factoring 722 is a lot easier if you’ve memorized that 19² = 361.

Here is today’s factoring puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here are some other interesting facts about the number 722:

I learned from Stetson.edu that (2^4) + (3^4) + (5^4) = 722. The red numbers are the first 3 prime numbers.

722 is also the sum of consecutive numbers two different ways:

• 179 + 180 + 181 + 182 = 722; that’s 4 consecutive numbers.
• 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39+ 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 722; that’s 19 consecutive numbers.

722 is a palindrome in two other bases:

• 2D2 BASE 16 (D = 13 base 10); note that 2(16²) + 13(16) + 2(1) = 722.
• 242 BASE 18; note that 2(18²) + 4(18) + 2(1) = 722.

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