factor pairs

Let’s Make a Factor Cake for 2020

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We often celebrate special occasions with a cake!

Coincidentally, there is a method to find the prime factorization of a number that is called the cake method.

Let’s make a factor cake for the year 2020 to celebrate its arrival!
2020 Factor Cake

make science GIFs like this at MakeaGif
The factor cake shows that the prime factorization of 2020 is 2 × 2 × 5 × 101. We can write that more compactly: 2020 = 2² × 5 × 101.
In case you would like a still picture of the cake instead of the gif, here it is:
I will write more about the number 2020 before tomorrow. Enjoy saying good-bye to 2019 and getting ready for the new year!

1452 Poinsettia Plant Mystery

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Today’s Puzzle:

Merry Christmas, Everybody!

The poinsettia plant has a reputation for being poisonous, but it has never been a part of a whodunnit, and it never will. Poinsettias actually aren’t poisonous.

Multiplication tables might also have a reputation for being deadly, but they aren’t either, except maybe this one. Can you use logic to solve this puzzle without it killing you?

To solve the puzzle, you will need some multiplication facts that you probably DON’T have memorized. They can be found in the table below. Be careful! The more often a clue appears, the more trouble it can be:

Notice that the number 60 appears EIGHT times in that table. Lucky for you, it doesn’t appear even once as a clue in today’s puzzle!

Factors of 1452:

Now I’d like to factor the puzzle number, 1452. Here are a few facts about that number:

1 + 4 + 5 + 2 = 12, which is divisible by 3, so 1452 is divisible by 3.
1 – 4 + 5 – 2 = 0, which is divisible by 11, so 1452 is divisible by 11.

  • 1452 is a composite number.
  • Prime factorization: 1452 = 2 × 2 × 3 × 11 × 11, which can be written 1452 = 2² × 3 × 11²
  • 1452 has at least one exponent greater than 1 in its prime factorization so √1452 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1452 = (√484)(√3) = 22√3
  • 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 1452 has exactly 18 factors.
  • The factors of 1452 are outlined with their factor pair partners in the graphic below.

1452 Factor Tree:

To commemorate the season, here’s a factor tree for 1452:

More about the Number 1452:

1452 is the difference of two squares three different ways:
364² – 362² = 1452,
124² – 118² = 1452, and
44² – 22² = 1452.

Have a very happy holiday!

1451 Star of Wonder

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Today’s Puzzle:

An important part of Mathematics is noticing patterns. I love it when mathematicians ask students, “What do you notice? What do you wonder?”

Do those questions puzzle you? They are questions you can ponder as you gaze on this star of wonder made from several different graphs.

 

To help distinguish the graphs, the dotted lines are exponential functions, the dashed lines are natural logarithm functions, and the solid lines are linear functions.

What do you notice? What do you wonder?

Factors of 1451:

Now I’ll tell you a little bit about the post number, 1451:

  • 1451 is a prime number.
  • Prime factorization: 1451 is prime.
  • 1451 has no exponents greater than 1 in its prime factorization, so √1451 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1451 has exactly 2 factors.
  • The factors of 1451 are outlined with their factor pair partners in the graphic below.

How do we know that 1451 is a prime number? If 1451 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1451. Since 1451 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1451 is a prime number.

More about the Number 1451:

1451 is the sum of two consecutive numbers:
726 + 725 = 1451.

1451 is also the difference of two consecutive squares:
726² – 725² = 1451.

What do you notice about those two facts?

1450 A Pair of Factor Trees

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On today’s puzzle, there are two small Christmas trees. Will two smaller trees on the puzzle be easier to solve than one big one? You’ll have to try it to know!

Every puzzle has a puzzle number to distinguish it from the others. Here are some facts about this puzzle number, 1450:

  • 1450 is a composite number.
  • Prime factorization: 1450 = 2 × 5 × 5 × 29, which can be written 1450 = 2 × 5² × 29
  • 1450 has at least one exponent greater than 1 in its prime factorization so √1450 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1450 = (√25)(√58) = 5√58
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1450 has exactly 12 factors.
  • The factors of 1450 are outlined with their factor pair partners in the graphic below.

In case you are looking for factor trees for 1450, here are two different ones:

1450 is the hypotenuse of SEVEN Pythagorean triples:
170-1440-1450 which is 10 times (17-144-145)
240-1430-1450 which is 10 times (24-143-145)
406-1392-1450 which is (7-24-25) times 58
666-1288-1450 which is 2 times (333-644-725)
728-1254-1450 which is 2 times (364-627-725)
870-1160-1450 which is (3-4-5) times 290
1000-1050-1450 which is (20-21-29) times 50

1449 Christmas Star

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If you’ve ever wished you knew the multiplication table better, then make that wish upon this Christmas star. If you use logic and don’t give up,  then you can watch your wish come true!

I number the puzzles to distinguish them from one another. That star puzzle is way too big for a factor tree made with its puzzle number:

Here’s more about the number 1449:

  • Prime factorization: 1449 = 3 × 3 × 7 × 23, which can be written 1449 = 3² × 7 × 23
  • 1449 has at least one exponent greater than 1 in its prime factorization so √1449 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1449 = (√9)(√161) = 3√161
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1449 has exactly 12 factors.
  • The factors of 1449 are outlined with their factor pair partners in the graphic below.

1449 is the difference of two squares in 6 different ways:
725² – 724² = 1449
243² – 240² = 1449
107²-100² = 1449
85² – 76² = 1449
45² – 24² = 1449
43² – 20² = 1449

1448 Christmas Factor Tree

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Here’s a puzzle that looks a little like a Christmas tree. Some of the clues might give you a little bit of trouble. For example, the common factor of 60 and 30 might be 5, 6, or 10. Likewise, the common factor of 8 and 4 might be 1, 2, or 4.

Which factor should you use? Look at all the other clues and use logic. Logic can help you write each of the numbers 1 to 12 in both the first column and the top row so that the given clues and those numbers behave like a multiplication table. Good luck!

I have to number every puzzle. It won’t help you solve the puzzle, but here are some facts about the number 1448:

The number made by its last two digits, 48, is divisible by 4, so 1448 is also divisible by 4. That fact can give us the first couple of branches of 1448’s factor tree:

  • 1448 is a composite number.
  • Prime factorization: 1448 = 2 × 2 × 2 × 181, which can be written 1448 = 2³ × 181
  • 1448 has at least one exponent greater than 1 in its prime factorization so √1448 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1448 = (√4)(√362) = 2√362
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1448 has exactly 8 factors.
  • The factors of 1448 are outlined with their factor pair partners in the graphic below.

1448 is also the hypotenuse of a Pythagorean triple:
152-1440-1448 which is 8 times (19-180-181)

1447 Christmas Light Puzzle

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If you’ve ever had a string of lights go out because ONE bulb went bad, it can be a very frustrating puzzle to figure out which light is causing the problem.

This is not that kind of puzzle. For this one, you just need to figure out where to put the numbers from 1 to 12 in both the first column and the top row so that the given clues are the products of those numbers. There is only one solution, and if you always use logic, it will not be a frustrating puzzle to solve.

I gave that puzzle the puzzle number 1447. That number won’t help you solve the puzzle, but here are some facts about it anyway:

  • 1447 is a prime number.
  • Prime factorization: 1447 is prime.
  • 1447 has no exponents greater than 1 in its prime factorization, so √1447 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1447 has exactly 2 factors.
  • The factors of 1447 are outlined with their factor pair partners in the graphic below.

How do we know that 1447 is a prime number? If 1447 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1447. Since 1447 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1447 is a prime number.

1447 is also the difference of two consecutive squares:
724² – 723² = 1447

 

 

 

1446 Peppermint Stick

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Red and green striped peppermint sticks are often seen in stores and homes in December. Can you lick this peppermint stick puzzle or will you let it lick you?

The puzzle number was 1446. Here are a few facts about that number:

  • 1446 is a composite number.
  • Prime factorization: 1446 = 2 × 3 × 241
  • 1446 has no exponents greater than 1 in its prime factorization, so √1446 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1446 has exactly 8 factors.
  • The factors of 1446 are outlined with their factor pair partners in the graphic below.

1446 is also the hypotenuse of a Pythagorean triple:
720-1254-1446 which is 6 times (120-209-241)

1445 Virgács for Your Boots Tonight

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Tomorrow is Mikulás (Saint Nicholas Day) in Hungary. Children will awake to find candy, fruit, or nuts in their polished shoes or boots because every boy and every girl has been at least a little bit good all year long.

Because they have also been at least a little bit naughty, they will also find virgács in those same shoes or boots. Virgács are little twigs that have been spray-painted gold and tied together at the top with red ribbon.

Santa is so busy this time of year, that I thought I would give him a helping hand. I’ve made some virgács for YOUR boots or shoes!

Start at the top of the puzzle and work your way down cell by cell to solve this Level 3 puzzle. Oh, but I’ve been just a little bit naughty making this puzzle: you will need to look at later clues to figure out what factors to give to 40. Will clue 40 use a 5 or a 10? Look at clues 60 and 90, and you will have only one choice for that answer. Then you can forgive my tiny bit of naughtiness.

Now I’ll tell you a few facts about the puzzle number, 1445:

  • 1445 is a composite number.
  • Prime factorization: 1445 = 5 × 17 × 17, which can be written 1445 = 5 × 17²
  • 1445 has at least one exponent greater than 1 in its prime factorization so √1445 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1445 = (√289)(√5) = 17√5
  • The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 1445 has exactly 6 factors.
  • The factors of 1445 are outlined with their factor pair partners in the graphic below.

1445 is the sum of two squares in THREE different ways:
31² + 22² = 1445
34² + 17² = 1445
38² + 1² = 1445

1445 is the hypotenuse of SEVEN Pythagorean triples:
76-1443-1445 calculated from 2(38)(1), 38² – 1², 38² + 1²
221-1428-1445 which is 17 times (13-84-85)
477-1364-1445 calculated from 31² – 22², 2(31)(22), 31² + 22²
612-1309-1445 which is 17 times (36-77-85)
680-1275-1445 which is (8-15-17) times 85
805-1200-1445 which is 5 times (161-240-289)
867-1156-1445 which is (3-4-5) times 289 and can
also be calculated from 34² – 17², 2(34)(17), 34² + 17²

1444 Christmas Wrapping Paper

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A fourth of the products in this multiplication table puzzle are already there just because I wanted the puzzle to have a wrapping paper pattern. Can you figure out what the factors are supposed to be and what all the other products are?

  • 1444 is a composite number and a perfect square.
  • Prime factorization: 1444 = 2 × 2 × 19 × 19, which can be written 1444 = 2²× 19²
  • 1444 has at least one exponent greater than 1 in its prime factorization so √1444 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1444 = (√4)(√361) = (2)(19) = 38
  • The exponents in the prime factorization are 2 and 2. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1444 has exactly 9 factors.
  • The factors of 1444 are outlined with their factor pair partners in the graphic below.

Square number 1444 looks a lot like another square number, 144.
If we keep adding 4’s to the end, will we continue to get square numbers?
No.

However, in different bases, 1444 looks like several other square numbers:
It’s 484 in BASE 18,
400 in BASE 19,
169 in BASE 35,
144 in BASE 36,
121 in BASE 37, and
100 in BASE 38.