A Multiplication Based Logic Puzzle

Posts tagged ‘factor tree’

960 Factor Trees

960 is the smallest number to have exactly 28 factors. 960 is 2⁶·3·5, so any factor tree made for it will have 6 + 1 + 1 = 8 prime factors. Since 8 is a power of 2, this number, 960, has some beautiful and well-balanced factor trees as well as some that aren’t as good-looking. Here are five of the MANY possible factor trees for 960:

 

960 can be written as the difference of 2 squares TEN different ways:

  1. 241² – 239² = (241 + 239)(241 – 239) = 480 × 2 = 960
  2. 122² – 118² = (122 + 118)(122 – 118) = 240 × 4 = 960
  3. 83² – 77² = (83 + 77)(83 – 77) = 160 × 6 = 960
  4. 64² – 56² = (64 + 56)(64 – 56) = 120 × 8 = 960
  5. 53² – 43² = (53 + 43)(53 – 43) = 96 × 10 = 960
  6. 46² – 34² = (46 + 34)(46 – 34) = 80 × 12 = 960
  7. 38² – 22² = (38 + 22)(38 – 22) = 60 × 16 = 960
  8. 34² – 14² = (34 + 14)(34 – 14) = 48 × 20 = 960
  9. 32² – 8² = (32 + 8)(32 – 8) = 40 × 24 = 960
  10. 31² – 1² = (31 + 1)(31 – 1) = 32 × 30 = 960

960 is the sum of the sixteen prime numbers from 29 to 97.

It is also the sum of six consecutive prime numbers:
149 + 151 + 157 + 163 + 167 + 173 = 960

960 is the hypotenuse of a Pythagorean triple:
576-768-960 which is (3-4-5) times 192

I like how 960 looks in these other bases:
33000 in BASE 4 because 3(4⁴) + 3(4³) = 3(256 + 64) = 3 × 320 = 960
440 in BASE 15 because 4(15²) + 4(15) = 4(225 + 15) = 4 × 240 = 960
UU in BASE 31 (U is 30 base 10), because 30(31) + 30(1) = 30(31 + 1) = 30 × 32 = 960
U0 in BASE 32 because 30(32) + 0 = 960

Stetson.edu informs us that 9 + 6 + 09³ + 6³ + 0³ = 960

  • 960 is a composite number and a perfect square.
  • Prime factorization: 960 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5, which can be written 960 = 2⁶ × 3 × 5
  • The exponents in the prime factorization are 6, 1 and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 960 has exactly 28 factors.
  • Factors of 960: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960
  • Factor pairs: 960 = 1 × 960, 2 × 480, 3 × 320, 4 × 240, 5 × 192, 6 × 160, 8 × 120, 10 × 96, 12 × 80, 15 × 64, 16 × 60, 20 × 48, 24 × 40, or 30 × 32
  • Taking the factor pair with the largest square number factor, we get √960 = (√64)(√15) = 8√15 ≈ 30.9838668

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864 Factor Trees

The prime factorization of 864 is 2⁵ × 3³. The sum of the exponents is 5 + 3 = 8. Since 8 is a power of 2,  a couple of 864’s factor trees are full and well-balanced:

All of those prime factors lined up in numerical order. That didn’t happen for the next one, but it still makes a good looking tree, and all the prime factors are easy to find.

Is it possible to make a factor tree for 864 that hardly looks like a tree and isn’t as easy to find all the prime factors? Yes, it is. Here’s an example:

864 looked interesting to me in a few other bases:

  • 4000 BASE 6 because 4(6³) = 864
  • 600 BASE 12 because 6(12²) = 864
  • RR BASE 31 (R is 27 base 10) because 27(31) + 27(1) = 27(32) = 864
  • OO BASE 35 (O is 24 base 10) because 24(35) + 24(1) = 24(36) = 864
  • O0 BASE 36 (Oh zero) because 24(36) + 0(1) = 864

864 is the sum of the 20 prime numbers from 7 to 83.

131 + 137 + 139 + 149 + 151 + 157 = 864; that’s six consecutive primes.

431 + 433 = 864; that’s the sum of twin primes.

  • 864 is a composite number.
  • Prime factorization: 864 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which can be written 864 = 2⁵ × 3³
  • The exponents in the prime factorization are 5 and 2. Adding one to each and multiplying we get (5 + 1)(3 + 1) = 6 × 4 = 24. Therefore 864 has exactly 24 factors.
  • Factors of 864: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 96, 108, 144, 216, 288, 432, 864
  • Factor pairs: 864 = 1 × 864, 2 × 432, 3 × 288, 4 × 216, 6 × 144, 8 × 108, 9 × 96, 12 × 72, 16 × 54, 18 × 48, 24 × 36, or 27 × 32
  • Taking the factor pair with the largest square number factor, we get √864 = (√144)(√6) = 12√6 ≈ 29.3938769

 

756 and Level 3

  • 756 is a composite number.
  • Prime factorization: 756 = 2 x 2 x 3 x 3 x 3 x 7, which can be written 756 = (2^2) x (3^3) x 7
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 756 has exactly 24 factors.
  • Factors of 756: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756
  • Factor pairs: 756 = 1 x 756, 2 x 378, 3 x 252, 4 x 189, 6 x 126, 7 x 108, 9 x 84, 12 x 63, 14 x 54, 18 x 42, 21 x 36 or 27 x 28
  • Taking the factor pair with the largest square number factor, we get √756 = (√21)(√36) = 6√21 ≈ 27.495454.

756-factor-pairs

 

756 has many factors and, therefore, it has many possible factor trees. Here are three of them:

756 factor trees

Here’s a level 3 Find the Factors puzzle for you to solve, too:

756 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s a few more thoughts on the number 756:

The last two digits of 756 is divisible by 4 so 756 is divisible by 4.

756 is formed from 3 consecutive numbers (5, 6, 7) so it is divisible by 3. The middle number is divisible by 3 so 756 is also divisible by 9.

756 can be written as the sum of consecutive numbers seven ways:

  • 251 + 252 + 253 = 756; that’s 3 consecutive numbers.
  • 105 + 106 + 107 + 108 + 109 + 110 + 111 = 756; that’s 7 consecutive numbers.
  • 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 = 756; that’s 8 consecutive numbers.
  • 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 756; that’s 9 consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 756; that’s 21 consecutive numbers.
  • 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 = 756; that’s 24 consecutive numbers.
  • 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 + 41  = 756; that’s 27 consecutive numbers.

756 is also the sum of six consecutive prime numbers: 109 + 113 + 127+ 131 + 137+ 139 = 756.

756 can be written as the sum of three squares four different ways. (Notice that all of the squares are even):

  • 26² + 8² + 4² = 756
  • 24² + 12² + 6² = 756
  • 22² + 16² + 4² = 756
  • 20² + 16² + 10² = 756

756 is a palindrome in two other bases:

  • 11011 BASE 5; note that 1(625) + 1(125) + 0(25) + 1(5) + 1(1) = 756.
  • LL BASE 35 (L is 21 base 10); note that 21(35) + 21(1) = 756.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

756 Factors

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 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:

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 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:

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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.

728-factor-pairs

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720 Christmas Factor Trees

720 has 30 factors. That’s more than any previous number!

  • 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.

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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688 is a Friedman number

688 is a palindrome in two different bases:

  • 2002 in base 7; note that 2(343) + 0(49) + 0(7) + 2(1) = 688
  • 494 in base 12; note that 4(144) + 9(12) + 4(1) = 688

688 is called a Friedman number because it can be expressed using only its own digits and +, -, x, ÷, ( ), and exponents. Stetson.edu has published a table of all such numbers that have 4 digits or less and the reason each qualifies to be a Friedman number.

688 is a Friedman number because 688 = 8 x 86 so I made a factor tree based on that single multiplication fact:

688 Factor Tree

Since it’s such a fun number fact, I positioned it on top of today’s factoring puzzle, too.

688 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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  • 688 is a composite number.
  • Prime factorization: 688 = 2 x 2 x 2 x 2 x 43, which can be written 688 = (2^4) x 43
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 688 has exactly 10 factors.
  • Factors of 688: 1, 2, 4, 8, 16, 43, 86, 172, 344, 688
  • Factor pairs: 688 = 1 x 688, 2 x 344, 4 x 172, 8 x 86, or 16 x 43
  • Taking the factor pair with the largest square number factor, we get √688 = (√16)(√43) = 4√43 ≈ 26.229754.

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688 Factors

330 Christmas Factor Trees

  • 330 is a composite number.
  • Prime factorization: 330 = 2 x 3 x 5 x 11
  • 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 330 has exactly 16 factors.
  • Factors of 330: 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330
  • Factor pairs: 330 = 1 x 330, 2 x 165, 3 x 110, 5 x 66, 6 x 55, 10 x 33, 11 x 30, or 15 x 22
  • 330 has no square factors that allow its square root to be simplified. √330 ≈ 18.166

Within these seven factor trees for 330 there are also factor trees for 6, 10, 15, 22, 30, 33, 55, 66, 110, and 165, the tops of which are all in brown. The prime factors of 330 are all in red.

330 Factor Trees

 

Can you find the factors and complete this Christmas tree multiplication table?2014-51 Level 2

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

2014-51 Level 2 Factors

 

 

 

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