# 664 The Back of a Halloween Cat’s Head

664  in base 12 is the palindrome 474. Note that 4(144) + 7(12) + 4(1)= 664.

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

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• 664 is a composite number.
• Prime factorization: 664 = 2 x 2 x 2 x 83, which can be written 664 = (2^3) x 83
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 664 has exactly 8 factors.
• Factors of 664: 1, 2, 4, 8, 83, 166, 332, 664
• Factor pairs: 664 = 1 x 664, 2 x 332, 4 x 166, or 8 x 83
• Taking the factor pair with the largest square number factor, we get √664 = (√4)(√166) = 2√166 ≈ 25.768197.

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Ricardo tweeted the solution:

# 663 Peanut Butter Kiss Puzzle

It’s easy to see that 663 is divisible by three.

That may not be very interesting at all, but…..

Because 13 and 17, 663’s other two prime factors, have a remainder of one when each is divided by four, 663 is the hypotenuse of FOUR Pythagorean triples. Can you find the greatest common factor of each triple?

• 420-513-663
• 63-660-663
• 255-612-663
• 312-585-663

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Art is not one of my strongest skills, but here is my attempt to make a puzzle that looks like a peanut butter kiss, a traditional Halloween candy:

Color can be inviting, but it can also be distracting. Here is the same puzzle without the added color:

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

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• 663 is a composite number.
• Prime factorization: 663 = 3 x 13 x 17
• 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 663 has exactly 8 factors.
• Factors of 663: 1, 3, 13, 17, 39, 51, 221, 663
• Factor pairs: 663 = 1 x 663, 3 x 221, 13 x 51, or 17 x 39
• 663 has no square factors that allow its square root to be simplified. √663 ≈ 25.748786.

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Thank you for sharing the solution, Ricardo, and happy Halloween:

# 662 More Candy Corn

662 is the sum of the twelve prime numbers from 31 to 79.

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

Here’s the same candy corn puzzle but less colorful.

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• 662 is a composite number.
• Prime factorization: 662 = 2 x 331
• 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 662 has exactly 4 factors.
• Factors of 662: 1, 2, 331, 662
• Factor pairs: 662 = 1 x 662 or 2 x 331
• 662 has no square factors that allow its square root to be simplified. √662 ≈ 25.72936.

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Ricardo tweeted the solution to the puzzle so I’m including it here as well.

# 661 Candy Corn

25² + 6² = 661

661 is the hypotenuse of the primitive Pythagorean triple 300-589-661 which was calculated using 2(25)(6), 25² – 6², 25² + 6².

661 is also the sum of the three prime numbers from 211 to 227. What is the prime number in the middle of the sum?

This Find the Factors puzzle is supposed to look like a piece of candy corn. 🙂

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

Here’s the puzzle without the possibly distracting color:

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• 661 is a prime number. 659 and 661 are twin primes.
• Prime factorization: 661 is prime.
• The exponent of prime number 661 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 661 has exactly 2 factors.
• Factors of 661: 1, 661
• Factor pairs: 661 = 1 x 661
• 661 has no square factors that allow its square root to be simplified. √661 ≈ 25.70992.

How do we know that 661 is a prime number? If 661 were not a prime number, then it would be divisible by at least one prime number less than or equal to √661 ≈ 25.7. Since 661 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 661 is a prime number.

Here’s another way we know that 661 is a prime number: Since 25² + 6² = 661, and 25 and 6 have no common prime factors, 661 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √661 ≈ 25.7. Since 661 is not divisible by 5, 13, or 17, we know that 661 is a prime number.

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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.

# 660 My Two-Year Blogiversary

Today is my 2 year blogiversary and my 660th post. Thank you to all my readers and followers. I really appreciate all of you.

What better way is there to celebrate than with a gorgeous cake? Enjoy!

The cake method of finding prime factors makes a beautiful cake for the number 660 because 660 has several prime factors and the largest one, 11, looks like a couple of candles to top it off perfectly.

660 has a lot of factors, 24 in fact. 660 is a special number for several reasons:

No number less than 660 has more factors than it does, but 360, 420, 480, 504, 540, 600, and 630 each have just as many.

660 has so many factors that it seems natural for it to between twin primes, 659 and 661.

660 is the hypotenuse of the Pythagorean triple 396-528-660. What is the greatest common factor of those three numbers?

660 is the sum of consecutive prime numbers three different ways. Prime number 101 is in two of those ways:

• 157 + 163 + 167 + 173 = 660
• 101 + 103 + 107 + 109 + 113 + 127 = 660
• 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 660

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• 660 is a composite number.
• Prime factorization: 660 = 2 x 2 x 3 x 5 x 11, which can be written 660 = 2² x 3 x 5 x 11
• 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) = 3 x 2 x 2 x 2 = 24. Therefore 660 has exactly 24 factors.
• Factors of 660: 1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330, 660
• Factor pairs: 660 = 1 x 660, 2 x 330, 3 x 220, 4 x 165, 5 x 132, 6 x 110, 10 x 66, 11 x 60, 12 x 55, 15 x 44, 20 x 33, or 22 x 30
• Taking the factor pair with the largest square number factor, we get √660 = (√4)(√165) = 2√165 ≈ 25.690465.

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# 659 Jack O’lantern Puzzle

659 is the sum of the 7 prime numbers from 79 to 107. Can you list them all before you add them up?

Enjoy this Jack O’lantern puzzle.

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

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• 659 is a prime number.
• Prime factorization: 659 is prime.
• The exponent of prime number 659 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 659 has exactly 2 factors.
• Factors of 659: 1, 659
• Factor pairs: 659 = 1 x 659
• 659 has no square factors that allow its square root to be simplified. √659 ≈ 25.670995.

How do we know that 659 is a prime number? If 659 were not a prime number, then it would be divisible by at least one prime number less than or equal to √659 ≈ 25.7. Since 659 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 659 is a prime number.

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Since riccardo took the time to photograph the puzzle’s solution, I decided to include it in this post as well. Great job riccardo!

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

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:

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:

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

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?

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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# 656 Halloween Tangram Puzzle Cats

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 puzzles are made from the seven shapes in this 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.
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:
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

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?

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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