A Multiplication Based Logic Puzzle

Posts tagged ‘factor pairs’

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.

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

 

Advertisements

986 and Level 1

Today’s puzzle looks like a simple holiday napkin, and it’s actually quite simple to solve. Even kids who have learned how to multiply but haven’t even heard of division could solve this puzzle.

986 looks the same upside-down as it does right-side up, so it is a strobogrammatic number.

31² + 5² = 986
25² + 19² = 986

986 is the hypotenuse of FOUR Pythagorean triples:
264-950-986
310-936-986
680-714-986
464-870-986

986 looks interesting in some other bases:
12421 BASE 5
TT in BASE 33 (T is 29 base 10)
T0 in BASE 34

  • 986 is a composite number.
  • Prime factorization: 986 = 2 × 17 × 29
  • 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 986 has exactly 8 factors.
  • Factors of 986: 1, 2, 17, 29, 34, 58, 493, 986
  • Factor pairs: 986 = 1 × 986, 2 × 493, 17 × 58, or 29 × 34
  • 986 has no square factors that allow its square root to be simplified. √986 ≈ 31.4006

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

982 Red-Hot Cinnamon Candy

When I was a child I remember eating a red-hot cinnamon ball around the holidays. I really like cinnamon, but I wasn’t sure I liked how hot the candy was. I hope you enjoy today’s red-hot cinnamon candy puzzle.

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

Now here’s something interesting about the number 982:

It is palindrome 292 in BASE 20 because 2(20²) + 9(20) + 2(1) = 982.

  • 982 is a composite number.
  • Prime factorization: 982 = 2 × 491
  • 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 982 has exactly 4 factors.
  • Factors of 982: 1, 2, 491, 982
  • Factor pairs: 982 = 1 × 982 or 2 × 491
  • 982 has no square factors that allow its square root to be simplified. √982 ≈ 31.336879

 

981 Peppermint Sticks

This time of year you can buy peppermint sticks that don’t just have red stripes, but they might have green ones, too. Today’s puzzle looks like a couple of peppermint sticks. It will be a sweet experience for you to solve it, so be sure to give it a try.

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

Here is some information about the number 981:

30² + 9² = 981 making 981 the hypotenuse of a Pythagorean triple:
540-819-981 which is 9 times (60-91-109) but can also be calculated from
2(30)(9), 30² – 9², 30² + 9²

981 is palindrome 171 in BASE 28 because 1(28²) + 7(28) + 1(1) = 981

  • 981 is a composite number.
  • Prime factorization: 981 = 3 × 3 × 109, which can be written 981 = 3² × 109
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 981 has exactly 6 factors.
  • Factors of 981: 1, 3, 9, 109, 327, 981
  • Factor pairs: 981 = 1 × 981, 3 × 327, or 9 × 109
  • Taking the factor pair with the largest square number factor, we get √981 = (√9)(√109) = 3√109 ≈ 31.3209

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.

Tag Cloud