factor pairs

658 How Many Triangles Point Up? How Many Triangles Point Down? How Many Triangles in All?

/ ivasallay / 3 Comments

658 Thirteen Rows of Triangles

When I first started counting the triangles in the figure, I started by counting the small triangles:

  • In the top row there is 1 small triangle.
  • In the top 2 rows there are 4 small triangles.
  • In the top 3 rows there are 9 small triangles.
  • In the top 4  rows there are 16 small triangles,
  • and so forth so that in all 13 rows there are 169 small triangles.

As fascinating as that resulting squaring pattern is, it is NOT part of the most efficient way to count ALL the triangles of varying sizes.

The most efficient way to count ALL the triangles can be found on The University of Georgia’s website: Count the triangles that are pointing up separately from the triangles pointing down. Counting charts for triangles with 4, 5, 6, 7, and 8 rows of triangles are displayed on that website. I made a similar chart for these 13 rows of triangles:

658 Chart, Triangles pointed up, Triangles pointed down

Notice all the triangular numbers on the chart!

Because 13 divided by 2 is 6.5, we see that 6 is the largest base size that has any triangles pointing down. Because 6.5 is not a whole number, there are 3 triangles that point down with a base size of 6.

Triangles made from an odd number of rows use these triangular numbers to count the triangles pointing down: 3, 10, 21, 36, 55, 78, etc.

Triangles made from an even number of rows have no remainder when divided by 2 so the triangle with the largest base size is the number of rows divided by two. There will only be one triangle with that base size and the triangular numbers used for that base size and smaller are 1, 6, 15, 28, 45, 66, 91, etc.

The Mathematics Stack Exchange had a discussion on how to count all the triangles, and a formula was posted:

The total number of triangles = ⌊n(n+2)(2n+1)/8⌋ Note: the brackets mean round decimals DOWN to the closest integer.

I made a chart showing the results of using the formula for n = 1 to 13:

658 Formula for number of triangles

Thus we see that 658 is the total number of triangles that can be counted in a triangle made from 13 rows of triangles.

  • 658 is a composite number.
  • Prime factorization: 658 = 2 x 7 x 47
  • 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 658 has exactly 8 factors.
  • Factors of 658: 1, 2, 7, 14, 47, 94, 329, 658
  • Factor pairs: 658 = 1 x 658, 2 x 329, 7 x 94, or 14 x 47
  • 658 has no square factors that allow its square root to be simplified. √658 ≈ 25.65151.

 

657 and Level 1

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657 is made from three consecutive numbers and the number in the middle, 6, is divisible by 3 so 657 is divisible by both 3 and 9.

657 is the hypotenuse of the Pythagorean triple 432-495-657. What is the greatest common factor of those three numbers?

657 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-26

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  • 657 is a composite number.
  • Prime factorization: 657 = 3 x 3 x 73, which can be written 657 = (3^2) x 73
  • 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 657 has exactly 6 factors.
  • Factors of 657: 1, 3, 9, 73, 219, 657
  • Factor pairs: 657 = 1 x 657, 3 x 219, or 9 x 73
  • Taking the factor pair with the largest square number factor, we get √657 = (√9)(√73) = 3√73 ≈ 25.632011.

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657 Factors

 

656 Halloween Tangram Puzzle Cats

/ ivasallay / 3 Comments

It’s almost Halloween, and I was in the mood to make some cats out of tangram pieces. MANY cat tangram patterns exist as well as patterns for witches hats, bats, haunted houses, and candles. Black makes the shapes look like shadows so I made a couple of patterns here (using excel).

tangram cats

Tangram puzzles are made from the seven shapes in this square:

labeled tangram square

How to fold and cut the tangram shapes from a half-sheet of black construction paper or other paper. (Thank you Paula Beardell Krieg from Bookzoompa.wordpress.com for the link on making a square and advice on the rest of the directions):

  1. First make a square from the rectangular piece of paper, and visualize that square becoming the labeled square above.
  2. Fold the square from the top left corner to the bottom right corner and crease the paper on the fold. Unfold the paper.
  3. Partially fold the square from the top right corner to the bottom left corner allowing only the left half of the square to show a crease. This crease and the previous crease will form the two large triangles.
  4. Fold the bottom right corner to the point where the other creases intersect. This new crease will form the medium triangle. Unfold. Cut out the medium triangle.tangrams minus medium triangle
  5. Refold the top right corner to the bottom left corner and crease the entire length this time.
  6. Cut along all the creases. You will then have cut out the two large triangles and be left with two odd pieces that look like this: tangram odd pieces
  7. Take the first odd piece and fold the longest straight edge so its endpoints meet. Crease and unfold. You should see a square and a small triangle. Cut them out.
  8. Place the newly cut small triangle over the second odd piece so that you see a small triangle and a parallelogram. Cut along the bottom edge of the small triangle so that you have a second small triangle and a parallelogram.

Now all seven pieces have been cut out and you can make your own cat or other fun creation!

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Here are some facts about the number 656:

656 is a palindrome in three bases:

  • 220022 in base 3; Note: 2(243) + 2(81) + 0(27) + 0(9) + 2(3) + 2(1) = 656
  • 808 in base 9; Note: 8(81) + 0(9) + 8(1) = 656
  • 656 in base 10; Obviously, 6(100) + 5(10) + 6(1) = 656

656 is the hypotenuse of the Pythagorean triple 144-640-656. What is the greatest common factor of those three numbers?

  • 656 is a composite number.
  • Prime factorization: 656 = 2 x 2 x 2 x 2 x 41, which can be written 656 = (2^4) x 41
  • 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 656 has exactly 10 factors.
  • Factors of 656: 1, 2, 4, 8, 16, 41, 82, 164, 328, 656
  • Factor pairs: 656 = 1 x 656, 2 x 328, 4 x 164, 8 x 82, or 16 x 41
  • Taking the factor pair with the largest square number factor, we get √656 = (√16)(√41) = 4√41 ≈ 25.612497.

655 and Level 6

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655  is the hypotenuse of the Pythagorean triple 393-524-655. What is the greatest common factor of those three numbers?

655 is a leg in exactly three triples. One of them is primitive; the rest are not. Which is which?

  • 655-1572-1703
  • 655-42900-42905
  • 655-214512-214513

Which of 655’s factors are the greatest common factors of each of the two that are not primitive?

655 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 655 is a composite number.
  • Prime factorization: 655 = 5 x 131
  • 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 655 has exactly 4 factors.
  • Factors of 655: 1, 5, 131, 655
  • Factor pairs: 655 = 1 x 655 or 5 x 131
  • 655 has no square factors that allow its square root to be simplified. √655 ≈ 25.592968.

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655 Logic

654 and Level 5

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654 is the sum of the fourteen prime numbers from 19 to 73.

654 is also the hypotenuse of the Pythagorean triple 360-546-654. Which of 654’s factors is the greatest common factor of those three numbers?

654 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 654 is a composite number.
  • Prime factorization: 654 = 2 x 3 x 109
  • 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 654 has exactly 8 factors.
  • Factors of 654: 1, 2, 3, 6, 109, 218, 327, 654
  • Factor pairs: 654 = 1 x 654, 2 x 327, 3 x 218, or 6 x 109
  • 654 has no square factors that allow its square root to be simplified. √654 ≈ 25.57342.

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654 Logic

652 and Level 3

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652 has 6 factors. 6 is a perfect number because all of its smaller factors, 1, 2, and 3, add up to its largest factor, 6.

The factors of 652 are 1, 2, 4, 163, and 326. The sum of those factors is 496, another perfect number. Note that all of 496’s smaller factors, 1, 2, 4, 8, 16, 31, 62, 124, and 248, add up to 496, its largest factor.

OEIS.org states that 652 is the only known non-perfect number that produces a perfect number in both of those situations.

652 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 652 is a composite number.
  • Prime factorization: 652 = 2 x 2 x 163, which can be written 652 = (2^2) x 163
  • 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 652 has exactly 6 factors.
  • Factors of 652: 1, 2, 4, 163, 326, 652
  • Factor pairs: 652 = 1 x 652, 2 x 326, or 4 x 163
  • Taking the factor pair with the largest square number factor, we get √652 = (√4)(√163) = 2√163 ≈ 25.53429.

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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.

652 Factors

651 and Level 2

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21 x 31 = 651. Both of those factors are 5 away from their average, 26, so 651 is 25 less than 26² or 676.

The numbers in each of 651’s four factor pairs are odd, and the average of each and the distance each is from that average are both whole numbers. That means that 651 can be expressed as the difference of two squares four different ways. In this particular case the averages and distances generate four primitive Pythagorean triples with 651 as one of the legs:

  • 26² – 5² = 651; primitive triple 260-651-701
  • 50² – 43² = 651; primitive triple 651-4300-4349
  • 110² – 107² = 651; primitive triple 651-23540-23549
  • 326² – 325² = 651; primitive triple 651-211900-211901

651 is also a  pentagonal number.

651 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 651 is a composite number.
  • Prime factorization: 651 = 3 x 7 x 31
  • 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 651 has exactly 8 factors.
  • Factors of 651: 1, 3, 7, 21, 31, 93, 217, 651
  • Factor pairs: 651 = 1 x 651, 3 x 217, 7 x 93, or 21 x 31
  • 651 has no square factors that allow its square root to be simplified. √651 ≈ 25.5147.

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651 Factors

650 is the sum of all the clues in this Level 1 puzzle

/ ivasallay / 2 Comments

1² + 2² + 3² + 4² + 5² + 6² +7² + 8² + 9² +10² + 11² + 12²  = 650

Thus 650 is the 12th square pyramidal number and can be calculated using 12(12 +1)(2⋅12 + 1)/6.

If you add up all the clues in today’s Find the Factors puzzle, you will get the number 650. However, if you print the puzzle from the excel file, one of the clues is missing because it isn’t needed to find the solution.

650 is the hypotenuse of seven Pythagorean triples!

  • 72-646-650
  • 160-630-650
  • 182-624-650
  • 250-600-650
  • 330-560-650
  • 390-520-650
  • 408-506-650

Can you find the greatest common factor of each triple? Each greatest common factor will be one of the factors of 650 listed below the puzzle.

650 is the hypotenuse of so many Pythagorean triples because it is divisible by 5, 13, 25, 65, and 325 which are also hypotenuses of triples. The smallest three numbers to be the hypotenuses of at least 7 triples are 325, 425, and 650.

Since 25 x 26 = 650, we know that (25-1)(26 + 1) = 650 – 2. Thus 24 x 27 = 648.

650 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 650 is a composite number.
  • Prime factorization: 650 = 2 x 5 x 5 x 13, which can be written 650 = 2 x (5^2) x 13
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 650 has exactly 12 factors.
  • Factors of 650: 1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650
  • Factor pairs: 650 = 1 x 650, 2 x 325, 5 x 130, 10 x 65, 13 x 50, or 25 x 26
  • Taking the factor pair with the largest square number factor, we get √650 = (√25)(√26) = 5√26 ≈ 25.495098.

650 Trees

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649 and Level 6

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6 – 4 + 9 = 11 so 649 is divisible by 11.

649 is the short leg in exactly three Pythagorean triples. Can you determine which one is a primitive triple, and what are the greatest common factors of each of the two non-primitive triples?

  • 649-3540-3599
  • 649-19140-19151
  • 649-210600-210601

649 Puzzle

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

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  • 649 is a composite number.
  • Prime factorization: 649 = 11 x 59
  • 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 649 has exactly 4 factors.
  • Factors of 649: 1, 11, 59, 649
  • Factor pairs: 649 = 1 x 649 or 11 x 59
  • 649 has no square factors that allow its square root to be simplified. √649 ≈ 25.475478.

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649 Logic

648 and Level 5

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648 is the sum of consecutive prime numbers 317 and 331.

The sixth root of 648 begins with 2.941682753. Notice all the digits from 1 to 9 appear somewhere in those nine decimal places. OEIS.org states that 648 is the smallest number that can make that claim.

From Archimedes-lab.org I learned some powerful facts about the number 648:

  • 16² – 17² + 18² – 19² + 20² – 21² +22² – 23² + 24² – 25² + 26² – 27² + 28² – 29² + 30² – 31² + 32² = 648
  • 48² – 47² + 46² – 45² + 44² – 43² +42² – 41² + 40² – 39² + 38² – 37² + 36² – 35² + 34² – 33² = 648
  • (1^2)(2^3)(3^4) = 648
  • 18² + 18²  = 648
  • (6^3) + (6^3) + (6^3) =648

648 Puzzle

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

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  • 648 is a composite number.
  • Prime factorization: 648 = 2 x 2 x 2 x 3 x 3 x 3 x 3, which can be written 648 = (2^3) x (3^4)
  • The exponents in the prime factorization are 3 and 4. Adding one to each and multiplying we get (3 + 1)(4 + 1) = 4 x 5 = 20. Therefore 648 has exactly 20 factors.
  • Factors of 648: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, 648
  • Factor pairs: 648 = 1 x 648, 2 x 324, 3 x 216, 4 x 162, 6 x 108, 8 x 81, 9 x 72, 12 x 54, 18 x 36, or 24 x 27
  • Taking the factor pair with the largest square number factor, we get √648 = (√324)(√2) = 18√2 ≈ 25.455844122…

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648 Logic