Month: January 2016

737 and Level 6

/ ivasallay / 3 Comments
  • 737 is a composite number.
  • Prime factorization: 737 = 11 x 67
  • 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 737 has exactly 4 factors.
  • Factors of 737: 1, 11, 67, 737
  • Factor pairs: 737 = 1 x 737 or 11 x 67
  • 737 has no square factors that allow its square root to be simplified. √737 ≈ 27.14774392.

737-factor-pairs

Here is a challenging level 6 puzzle. Some possible steps to solve it are in a table at the end of this post.

737 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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The Boeing 737 is the best selling commercial airline jet. It has been produced continuously since 1967. If you’ve ever flown Southwest Airlines, Ryanair, United Airlines, or American Airlines, chances are the 737 took you to your destination.

The next time you fly in a 737, use the following number facts to impress your seatmates. They will be so glad they sat next to you: 😉

7 – 3 + 7 = 11 so 737 is divisible by 11.

737 can be written as the sum of consecutive numbers three ways:

  • 368 + 369 = 737; that’s 2 consecutive numbers.
  • 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 = 737; that’s 11 consecutive numbers.
  • 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 = 737; that’s 22 consecutive numbers.

737 is obviously a palindrome, but base 10 is not the only base that can make that claim:

  • 737 base 10; note that 7(100) + 3(10) + 7(1) = 737.
  • 515 BASE 12; note that 5(144) + 1(12) + 5(1) = 737.
  • 191 BASE 23; note that 1(529) + 9(23) + 1(1) = 737.

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737 Logic

736 and Level 5

/ ivasallay / Leave a comment
  • 736 is a composite number.
  • Prime factorization: 736 = 2 x 2 x 2 x 2 x 2 x 23, which can be written 732 = (2^5) x 23
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 736 has exactly 12 factors.
  • Factors of 736: 1, 2, 4, 8, 16, 23, 32, 46, 92, 184, 368, 736
  • Factor pairs: 736 = 1 x 736, 2 x 368, 4 x 184, 8 x 92, 16 x 46, or 23 x 32
  • Taking the factor pair with the largest square number factor, we get √736 = (√16)(√46) = 4√46 ≈ 27.1293199.

736-factor-pairs

Some great online resources for teachers are on the Mathfireworks website. I am very pleased that Find the Factors made the list. It is included under Puzzles and Games. Help for solving today’s puzzle can be found in the table at the bottom of this post.

736 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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23 ⋅ 32 = 736 so it is the product of semordnilaps. (Semordnilap is palindromes spelled backwards, so semordnilap and palindromes are semordnilaps.)

736 = 7 + 3^6 making it the 14th Friedman number. Since 736 is equal to an expression that uses only “+ – × / ^ ( )”, all of its digits (or a concatenation of its digits) in the order in which they occur in the number, it is the 3rd nice Friedman number. Thank you OEIS.org for that fun number fact. (All of the suggested links are different and well worth a look.) Note: (736) = 736 is trivial so it would not satisfy the definition of a Friedman number.

21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 = 736; that’s 23 consecutive numbers.

735 and 736 are the smallest pair of consecutive numbers with 12 factors each.

736 is a palindrome in 2 bases:

  • 1E1 BASE 21 (E = 14 base 10); note that 1(21^2) + 14(21^1) + 1(21^0) = 736.
  • NN BASE 31 (N = 23 base 10); note that 23(31) + 23(1) = 736.

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736 Logic

735 and Level 4

/ ivasallay / Leave a comment
  • 735 is a composite number.
  • Prime factorization: 735 = 3 x 5 x 7 x 7, which can be written 735 = 3 x 5 x (7^2)
  • 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 735 has exactly 12 factors.
  • Factors of 735: 1, 3, 5, 7, 15, 21, 35, 49, 105, 147, 245, 735
  • Factor pairs: 735 = 1 x 735, 3 x 245, 5 x 147, 7 x 105, 15 x 49, or 21 x 35
  • Taking the factor pair with the largest square number factor, we get √735 = (√49)(√15) = 7√15 ≈ 27.11088.

735-factor-pairs

Here’s a medium level factoring puzzle for you to try:

735 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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735 = 7²·3·5. OEIS.org informs us that 735 is the smallest number whose digits and distinct prime factors are the same.

735 is the sum of consecutive numbers many different ways:

  • 367 + 368 = 735; that’s 2 consecutive numbers.
  • 244 + 245 + 246 = 735; that’s 3 consecutive numbers.
  • 145 + 146 + 147 + 148 + 149 = 735; that’s 5 consecutive numbers.
  • 120 + 121 + 122 + 123 + 124 + 125 = 735; that’s 6 consecutive numbers.
  • 102 + 103 + 104 + 105 + 106 + 107 + 108 = 735; that’s 7 consecutive numbers.
  • 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 = 735; that’s 10 consecutive numbers.
  • 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 = 735; that’s 14 consecutive numbers.
  • 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 = 735; that’s 15 consecutive numbers.
  • 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 735; that’s 21 consecutive numbers.
  • 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 = 735; that’s 30 consecutive numbers.
  • 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 = 735; that’s 35 consecutive numbers.

Because 5 is one of its factors, 735 is the hypotenuse of Pythagorean triple 441-588-735. What is the greatest common factor of those three numbers? 147, the other half of the factor pair with 5.

735 is a palindrome in two other bases:

  • 3223 BASE 6; note that 3(216) + 2(36) + 2(6) + 3(1) = 735.
  • LL BASE 34 (L = 21 base 10); note that 21(24) + 21(1) = 735.

If I were calculating the square root of 735, I would make a little factor cake using only prime numbers and prime numbers squared on the outside of the cake, and then I would take the square root of everything on the outside of the cake:

735 square root cake

 

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735 Logic

 

There Are 10 Ways 734 is the Sum of 3 Squares

/ ivasallay / 2 Comments
  • 734 is a composite number.
  • Prime factorization: 734 = 2 x 367
  • 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 734 has exactly 4 factors.
  • Factors of 734: 1, 2, 367, 734
  • Factor pairs: 734 = 1 x 734 or 2 x 367
  • 734 has no square factors that allow its square root to be simplified. √734 ≈ 27.092434.

734-factor-pairs

Here is a factoring puzzle for you to try. If you need some hints, scroll down the page.

734 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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Here’s a little more about the number 734:

182 + 183 + 184 + 185 = 734; that means 734 is the sum of 4 consecutive numbers.

OEIS.org informs us that 734 is the smallest number that can be expressed as the sum of 3 square numbers 10 different ways. I couldn’t resist the challenge of finding what those 10 ways are:

  1. 27² + 2² + 1² = 734
  2. 26² + 7² + 3² = 734
  3. 25² + 10² + 3² = 734
  4. 23² + 14² + 3² = 734
  5. 23² + 13² + 6² = 734
  6. 22² + 15² + 5² = 734
  7. 22² + 13² + 9² = 734
  8. 21² + 17² + 2² = 734
  9. 19² + 18² + 7² = 734
  10. 18² + 17² + 11² = 734

734 is palindrome 23132 in BASE 4; note that 2(4^4) + 3(4^3) + 1(4^2) + 3(4^1) + 2(4^0) = 734

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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 10.)  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.

734 Factors

733 and Level 2

/ ivasallay / Leave a comment
  • 733 is a prime number.
  • Prime factorization: 733 is prime.
  • The exponent of prime number 733 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 733 has exactly 2 factors.
  • Factors of 733: 1, 733
  • Factor pairs: 733 = 1 x 733
  • 733 has no square factors that allow its square root to be simplified. √733 ≈ 27.0739727.

How do we know that 733 is a prime number? If 733 were not a prime number, then it would be divisible by at least one prime number less than or equal to √733 ≈ 27.1. Since 733 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 733 is a prime number.

Here’s another way we know that 733 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 17² + 2² = 733 with 17 and 2 having no common prime factors, 733 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √733 ≈ 27.1. Since 733 is not divisible by 5, 13, or 17, we know that 733 is a prime number.

733-factor-pairs

Give this Level 2 factoring puzzle a try:

733 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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Here are some more facts about the number 733:

733 is prime, so 366 + 367 = 733 is the only way it can be written as the sum of consecutive numbers.

733 is the sum of the prime numbers from 17 to 79. I think there’s a good chance you know what all those prime numbers are.

733 is also the sum of the five prime numbers from 137  to 157. You might not know what the missing primes are, but here’s a hint: If you subtract 100 from any of those three primes, you will get a composite number, and there are only three odd composite numbers that don’t end in 5 between 37 and 57.

27² + 2² = 733 so 733 is the hypotenuse of the (primitive) Pythagorean triple 108-725-733 which is calculated from 2(27)(2), 27² – 2², 27² + 2².

Thus, 108² + 725² = 733².

733 is palindrome 292 in BASE 17; note that 2(17²) + 9(17) + 2(1) = 733.

From Wikipedia I learned two other interesting facts about the number 733:

  • 727, 733, 739 are consecutive prime numbers whose average is 733.  That makes 733 the 13th balanced prime. It is exactly the same distance from the previous prime and the prime that follows it.
  • 337, 373, 733 are three of only nine 3-digit permutable prime numbers. No matter their order, those exact 3 digits produce a prime number.

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733 Factors

732 and Level 1

/ ivasallay / Leave a comment
  • 732 is a composite number.
  • Prime factorization: 732 = 2 x 2 x 3 x 61, which can be written 732 = (2^2) x 3 x 61
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 732 has exactly 12 factors.
  • Factors of 732: 1, 2, 3, 4, 6, 12, 61, 122, 183, 244, 366, 732
  • Factor pairs: 732 = 1 x 732, 2 x 366, 3 x 244, 4 x 183, 6 x 122, or 12 x 61
  • Taking the factor pair with the largest square number factor, we get √732 = (√4)(√183) = 2√183 ≈ 27.0554985.

732-factor-pairs

Finding the factors that solve this factoring puzzle will be easier than that!

732 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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Here are some more facts about the number 732:

732 is the sum of consecutive numbers several different ways:

  • 243 + 244 + 245 = 732; that’s 3 consecutive numbers.
  • 88 + 89 + 90 + 91 + 92 + 93 + 94 + 95 = 732; that’s 8 consecutive numbers.
  • 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 732; that’s 24 consecutive numbers.

732 is the sum of the 8 prime numbers from 73 to 107. It is also the sum of the 10 prime numbers from 53 to 97. Do you know what all those prime numbers are?

Because 61 is one of its factors, 732 is the hypotenuse of Pythagorean triple 132-720-732. That means 132² + 720² = 732².

732 is a palindrome in two bases:

  • 606 BASE 11; note that 6(121) + 0(11) + 6(1) = 732.
  • 444 BASE 13; note that 4(169) + 4(13) + 4(1) = 732.

7^1 + 6^2 + 5^3 + 4^4 + 3^5 + 2^6 + 1^7 = 732. Thank you, OEIS.org for that fun number fact.

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732 Factors

731 and Level 6

/ ivasallay / Leave a comment
  • 731 is a composite number.
  • Prime factorization: 731 = 17 x 43
  • 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 731 has exactly 4 factors.
  • Factors of 731: 1, 17, 43, 731
  • Factor pairs: 731 = 1 x 731 or 17 x 43
  • 731 has no square factors that allow its square root to be simplified. √731 ≈ 27.03711669.

Here is a Level 6 puzzle for you try:

731 Puzzle

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

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731 can be written as the sum of consecutive numbers 3 different ways:

  • 365 + 366 = 731; that’s 2 consecutive numbers.
  • 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 = 731; that’s 17 consecutive numbers.
  • 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 = 731; that’s 34 consecutive numbers.

Because 17 is one of its factors,  731 is the hypotenuse of Pythagorean triple 344-645-731 making 344² + 645² = 731².

239 + 241 + 251 = 731 so 731 is the sum of 3 consecutive prime numbers.

731 is palindrome 3A3 in BASE 14; note that 3(14²) + 10(14) + 3(1) = 731.

731 is also 123 in BASE 26; note that 1(26²) + 2(26) + 3(1) = 731.

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731 Logic

Thanks to Ricardo whose tweet I include here:

//platform.twitter.com/widgets.js

730 and Level 5

/ ivasallay / Leave a comment

Two years have 730 days when neither year is a leap year.

  • 730 is a composite number.
  • Prime factorization: 730 = 2 x 5 x 73
  • 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 730 has exactly 8 factors.
  • Factors of 730: 1, 2, 5, 10, 73, 146, 365, 730
  • Factor pairs: 730 = 1 x 730, 2 x 365, 5 x 146, or 10 x 73
  • 730 has no square factors that allow its square root to be simplified. √730 ≈ 27.01851.

730-factor-pairs

Here are some statistics that WordPress gave me just before 2015 ended: findthefactors.com 2015annual report.

730 Puzzle

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

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Here a few more facts about the number 730:

730 is the sum of consecutive numbers several ways:

  • 181 + 182 + 183 + 184 = 730; that’s 4 consecutive numbers.
  • 144 + 145 + 146 + 147 + 148 = 730; that’s 5 consecutive numbers.
  • 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 730; that’s 20 consecutive numbers.

730 is also the sum of two squares two ways:

  • 27² + 1² = 730
  • 21² + 17² = 730

Because 5 and 73 are two of its factors, 730 is the hypotenuse of FOUR Pythagorean triples (none are primitive):

  • 54-728-730, calculated from 2(27)(1), 27² – 1², 27² + 1²
  • 152-714-730, calculated from 21² – 17², 2(21)(17), 21² + 17²
  • 438-584-730, and 146 is their GCF (greatest common factor).
  • 480-550-730, and 10 is their GCF.

730 is a palindrome in three bases:

  • 1000001 BASE 3
  • 1001 BASE 9
  • 101 BASE 27

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730 Logic

729 What I Did for Paula Krieg’s Star Project

/ ivasallay / Leave a comment

Let me tell you a little about 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:

  • 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.

729-factor-pairs