1332 Yet Another Christmas Tree

 

Here is yet another Christmas tree for you to enjoy this holiday season.

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Here are a few facts about the number 1332:

  • 1332 is a composite number.
  • Prime factorization: 1332 = 2 × 2 × 3 × 3 × 37, which can be written 1332 = 2² × 3² × 37
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1332 has exactly 18 factors.
  • Factors of 1332: 1, 2, 3, 4, 6, 9, 12, 18, 36, 37, 74, 111, 148, 222, 333, 444, 666, 1332
  • Factor pairs: 1332 = 1 × 1332, 2 × 666, 3 × 444, 4 × 333, 6 × 222, 9 × 148, 12 × 111, 18 × 74 or 36 × 37
  • Taking the factor pair with the largest square number factor, we get √1332 = (√36)(√37) = 6√37 ≈ 36.49658

Here are a couple of factor trees for 1332:

Since 36 × 37 = 1332, we know that 1332 is the sum of the first 36 even numbers. (The first 36 numbers add up to the infamous 666, and 2 times 666 is 1332.)

Because 1332 is divisible by both 3 and 37, it has several repdigits as factors, 111, 222, 333, 444, and 666.

1332 is the sum of four consecutive prime numbers:
317 + 331 + 337 + 347 = 1332

1332 is the sum of two square numbers:
36² + 6² = 1332

1332 is the hypotenuse of a Pythagorean triple:
432-1260-1332 which is 36 times (12-35-37)
It can also be calculated from 2(36)(6), 36² – 6², 36² + 6²

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1328 Christmas Tree

 

Can you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that the lights on this Christmas tree work properly?

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Now I’ll share some facts about the puzzle number, 1328:

  • 1328 is a composite number.
  • Prime factorization: 1328 = 2 × 2 × 2 × 2 × 83, which can be written 1328 = 2⁴ × 83
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1328 has exactly 10 factors.
  • Factors of 1328: 1, 2, 4, 8, 16, 83, 166, 332, 664, 1328
  • Factor pairs: 1328 = 1 × 1328, 2 × 664, 4 × 332, 8 × 166, or 16 × 83
  • Taking the factor pair with the largest square number factor, we get √1328 = (√16)(√83) = 4√83 ≈ 36.44173

 

Because 28 is divisible by 4, but not by 8, and 3 (the digit before the 28) is an odd number, I know that 1328 is divisible by 8. I can use that fact to make this simple factor tree:

1328 is the difference of two squares three different ways:
333² – 331² = 1328
168² – 164² = 1328
87² – 79²  = 1328

993 Christmas Angel

There are 21 clues in this Christmas Angel puzzle. Will it be easy or difficult for you to solve? That is part of the mystery. As always, there is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

When is 993 a palindrome?
It is 5B5 in BASE 13 (B is 11 base 10) because 5(169) + 11(13) + 5(1) = 993,
313 in BASE 18 because 3(18²) + 1(18) + 3(1) = 993,
and it is repdigit 111 in BASE 31 because 31² + 31¹ + 31⁰ = 993

  • 993 is a composite number.
  • Prime factorization: 993 = 3 × 331
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 993 has exactly 4 factors.
  • Factors of 993: 1, 3, 331, 993
  • Factor pairs: 993 = 1 × 993 or 3 × 331
  • 993 has no square factors that allow its square root to be simplified. √993 ≈ 31.5119

992 Christmas Factor Tree

Artificial Christmas trees have to be assembled. Sometimes the assembly is easy, and sometimes it is frustrating.

This Christmas tree puzzle can be solved using LOGIC and an ordinary multiplication table, but there’s a good chance it will frustrate you. Go ahead and try to solve it!

Print the puzzles or type the solution in this excel file: 10-factors-986-992

The number 992 also can make a nice looking, well-balanced factor tree:

992 is the product of two consecutive numbers: 31 × 32 = 992.
Because of that fact, 992 is the sum of the first 31 EVEN numbers:
2 + 4 + 6 + 8 + 10 + . . . + 54 + 56 + 58 + 60 + 62 = 992

992 is palindrome 212 in BASE 22 because 2(22²) + 1(22) + 2(1) = 922. That was a lot of 2’s and 1’s in that fun fact!

  • 992 is a composite number.
  • Prime factorization: 992 = 2 × 2 × 2 × 2 × 2 × 31, which can be written 992 = 2⁵ × 31
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 992 has exactly 12 factors.
  • Factors of 992: 1, 2, 4, 8, 16, 31, 32, 62, 124, 248, 496, 992
  • Factor pairs: 992 = 1 × 992, 2 × 496, 4 × 248, 8 × 124, 16 × 62, or 31 × 32
  • Taking the factor pair with the largest square number factor, we get √992 = (√16)(√62) = 4√62 ≈ 31.49603

989 Christmas Bells at Eventide

I made this puzzle with silver clues to look like a bell. I thought dark blue looked best as a background color with the silver numbers. The puzzle reminds me of a bell in the evening. For the fun of it, I googled Christmas Bells in the evening to see if any poems or songs came up.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

I was very surprised that Google found a very old song called Christmas Bells at Eventide. I did not know that such a song existed. Eventide means the same thing as evening. You can listen to the song below.

This is my 989th post.

The number 989 is a palindrome in base 10. What about any other bases?
It’s 373 in BASE 17 because 3(17²) + 7(17) + 3(1) = 989,
252 in BASE 21 because 2(21²) + 5(21) + 2(1) = 989, and
1C1 in BASE 26 (C is 12 base 10) because 1(26²) + 12(26) + 1(1) = 989

  • 989 is a composite number.
  • Prime factorization: 989 = 23 × 43
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 989 has exactly 4 factors.
  • Factors of 989: 1, 23, 43, 989
  • Factor pairs: 989 = 1 × 989 or 23 × 43
  • 989 has no square factors that allow its square root to be simplified. √989 ≈ 31.44837

987 Christmas Star

Today’s puzzle is a lovely Christmas star whose golden beams shine throughout the dark night. Solving this puzzle could also enlighten your mind.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

987 is made from three consecutive numbers so it is divisible by 3.

It is also the sixteen number in the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, . . .

987 is a palindrome when written in base 11 or base 29:
818 in BASE 11 because 8(121) + 1(11) + 8(1) = 987
151 in BASE 29 because 1(29²) + 5(29) + 1(1) = 987

  • 987 is a composite number.
  • Prime factorization: 987 = 3 × 7 × 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 × 2 × 2 = 8. Therefore 987 has exactly 8 factors.
  • Factors of 987: 1, 3, 7, 21, 47, 141, 329, 987
  • Factor pairs: 987 = 1 × 987, 3 × 329, 7 × 141, or 21 × 47
  • 987 has no square factors that allow its square root to be simplified. √987 ≈ 31.416556

 

985 Make an Origami Santa Star

Almost immediately when I saw Paula Beardell Krieg’s origami stars, I thought about turning one into a Santa Star. It occurred to me that the white pentagon formed on the back of the star would make a nice beard for Santa. I made a prototype and tweaked it and tweaked it until I got this result:

Why did I want to do this? My daughter-in-law, Michelle, adores Santa Stars. They are her favorite Christmas decoration. When I gave her this Santa Star, she got so excited. She recently took a picture of her collection, and I am thrilled that the one I made for her was included.

If you would like to make this Santa Star, follow these steps.

  1. Click on the pentagon above, then copy and paste it into a document. Make it as big as your printer allows.
  2. Print the pentagon and cut it out.
  3. Follow the directions in the video by Tobias that Paula recommends.

Here are some pictures I took as I folded mine. Click on them if you want to see them better. I’ve also included a few tips to help you in folding the star:

In this picture, you can see that the pentagon was folded in half five different ways in the first set of folds. The second set of folds creates a smaller pentagon in the center of the pentagon as well as a star-like shape.

The third set of folds creates a new crease. I make a flower-like shape by refolding that crease on each side. To me, this “flower” is a very important step to get the paper to form the star.

Those creases will help form the small white pentagon you see in the picture below that will become Santa’s beard.

Turn the paper over to reveal a bigger pentagon. You will fold the vertices of this pentagon to the center of the pentagon. Fold the red and black tips at the same time as you fold the vertices. After you make the first fold, I recommend unfolding it. Fold the other vertices in order so that first fold will eventually become your last fold. The last fold is the most difficult to do. If it has already been folded once, it will be much easier to fold at the end.

Again, here is the finished Santa star.

Now I’ll share some facts about the number 985:

985 is the sum of three consecutive prime numbers:
317 + 331 + 337 = 985

29² + 12² = 985 and 27² + 16² = 985

985 is the hypotenuse of FOUR Pythagorean triples
140-975-985
591-788-985
473-864-985
696-697-985

When is 985 a palindrome?
It’s 505 in BASE 14 because 5(14²) + 5(1) = 5(196 + 1) = 5(197) = 985
It’s 1H1 BASE in 24 (H is 17 base 10) because 1(24²) + 17(24) + 1(1) = 985

  • 985 is a composite number.
  • Prime factorization: 985 = 5 × 197
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 985 has exactly 4 factors.
  • Factors of 985: 1, 5, 197, 985
  • Factor pairs: 985 = 1 × 985 or 5 × 197
  • 985 has no square factors that allow its square root to be simplified. √985 ≈ 31.3847

984 Way Too Big Christmas Factor Tree

Some Christmas trees are so big they are difficult to take home in the car. They might even be too big to set up in the house. This puzzle is the biggest one I have ever made. It looks like a very big Christmas tree waiting to be set up. Is it too big to bring fun this Christmastime?

The table below may be helpful in solving the puzzle. There are 400 places to write products in a 20 × 20 multiplication table, but not all the numbers from 1 to 400 appear in such a table. Some numbers don’t appear at all while other numbers appear more than one times. The chart below is color-coded to show how many times a product appears in the 20 × 20 multiplication table. Clues in the puzzle that appear only once (yellow) or twice (green) in the multiplication table won’t cause much trouble when solving the puzzle. Any other clues might stump you. Notice that the number 60 appears twice in the puzzle but eight times (black) in the 20 × 20 multiplication table!

I’d like to share some information about the number 984.

It is the hypotenuse of a Pythagorean triple:
216-960-984 which is 24 times (9-40-41)

Stetson.edu informs us that 8 + 88 + 888 = 984.

984 is 1313 in BASE 9 because 1(9³) + 3(9²) + 1(9¹) + 3(9⁰) = 984.

Here are a couple of the many possible factor trees for 984:

  • 984 is a composite number.
  • Prime factorization: 984 = 2 × 2 × 2 × 3 × 41, which can be written 984 = 2³ × 3 × 41
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 984 has exactly 16 factors.
  • Factors of 984: 1, 2, 3, 4, 6, 8, 12, 24, 41, 82, 123, 164, 246, 328, 492, 984
  • Factor pairs: 984 = 1 × 984, 2 × 492, 3 × 328, 4 × 246, 6 × 164, 8 × 123, 12 × 82, or 24 × 41
  • Taking the factor pair with the largest square number factor, we get √984 = (√4)(√246) = 2√246 ≈ 31.36877

983 Candy Cane

Candy canes have been a part of the Christmas season for ages. Here’s a candy cane puzzle for you to try. It’s a level 6 so it won’t be easy, but you will taste its sweetness once you complete it. Go ahead and get started!

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Here’s some information about prime number 983:

983 is the sum of consecutive prime numbers two different ways:
It is the sum of the seventeen prime numbers from 23 to 97.
It is also the sum of the thirteen prime numbers from 47 to 103.

  • 983 is a prime number.
  • Prime factorization: 983 is prime.
  • The exponent of prime number 983 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 983 has exactly 2 factors.
  • Factors of 983: 1, 983
  • Factor pairs: 983 = 1 × 983
  • 983 has no square factors that allow its square root to be simplified. √983 ≈ 31.35283

How do we know that 983 is a prime number? If 983 were not a prime number, then it would be divisible by at least one prime number less than or equal to √983 ≈ 31.4. Since 983 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 983 is a prime number.

980 Christmas Factor Trees

This level 4 puzzle has 12 clues in it that are products of factor pairs in which both factors are numbers from 1 to 12. The clues make an evergreen tree, the symbol of everlasting life which is so fitting for Christmas. Can you find the factors for the given clues and put them in the right places?

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Now I’ll tell you a little about the number 980:

It has eighteen factors and many possible factor trees. Here are just three of them:

28² + 14² = 980, so 980 is the hypotenuse of a Pythagorean triple:
588-784-980 which is (3-4-5) times 196, but can also be calculated from
28² – 14², 2(28)(14), 28² + 14²

I like the way 980 looks in some other bases:
It is 5A5 in BASE 13 (A is 10 base 10) because 5(13) + 10(13) + 5(1) = 980,
500 in BASE 14 because 5(14²) = 980,
SS in BASE 34 (S is 28 base 10) because 28(34) + 28(1) = 28(35) = 980
S0 in BASE 35 because 28(35) = 980

  • 980 is a composite number.
  • Prime factorization: 980 = 2 × 2 × 5 × 7 × 7, which can be written 980 = 2² × 5 × 7²
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 980 has exactly 18 factors.
  • Factors of 980: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980
  • Factor pairs: 980 = 1 × 980, 2 × 490, 4 × 245, 5 × 196, 7 × 140, 10 × 98, 14 × 70, 20 × 49 or 28 × 35
  • Taking the factor pair with the largest square number factor, we get √980 = (√196)(√5) = 14√5 ≈ 31.30495.