Let’s Make a Factor Cake for 2020

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!
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1451 Star of Wonder

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

Those 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?

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.

1449 Christmas Star

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

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)

1445 Virgács for Your Boots Tonight

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

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 1, 1, and 1. 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.

1442 A Birthday Present for My Brother, Andy

My brother, Andy, has a birthday today. He’s very good at solving puzzles, so I made this Challenge puzzle as a  present for him.  Have a very happy birthday, Andy!

You can try to solve it too. If the box and ribbon are too distracting, here’s a copy of the puzzle without the added color. Click on it to see it better.

Print the puzzles or type the solution in this excel file: 10 Factors 1432-1442

That was puzzle number 1442. Here are some facts about that number:

  • 1442 is a composite number.
  • Prime factorization: 1442 = 2 × 7 × 103
  • 1442 has no exponents greater than 1 in its prime factorization, so √1442 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 1442 has exactly 8 factors.
  • The factors of 1442 are outlined with their factor pair partners in the graphic below.

2(103)(7) = 1442, making it a leg in this Pythagorean triple:
1442-10560-10658 calculated from 2(103)(7), 103² – 7², 103² + 7²

Why is 1441 a Star?

1441 is the 16th star number. If you look at the tiny squares that make up each triangle in the star, you might notice that each of those triangles is made with 15 rows of squares. Thus each triangle represents the 15th triangular number.

A six-pointed star is really just a 12-sided figure better known as a dodecagon.

Since I recently wrote that figurate numbers can be found using triangular numbers, I wondered if centered figurate numbers can make the same claim. Yes, they can!

Centered Figurate Numbers are also called Centered Polygonal Numbers and can be easily calculated from the triangular numbers.

Here are some more facts about the number 1441:

  • 1441 is a composite number.
  • Prime factorization: 1441 = 11 × 131
  • 1441 has no exponents greater than 1 in its prime factorization, so √1441 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1441 has exactly 4 factors.
  • The factors of 1441 are outlined with their factor pair partners in the graphic below.

1441 is a palindrome with an even number of digits so it is divisible by 11.

1441 is also a palindrome in some other bases, namely, base 6, base 30, base 32, and base 36.

Taking the Mystery out of 1440

The number 1440 has thirty-six factors. That’s a lot! The only number smaller than it with that many factors is 1260. Why do they have so many?

Would solving this Mystery Level puzzle be an easier mystery to solve?

Print the puzzles or type the solution in this excel file: 10 Factors 1432-1442

  • 1440 is a composite number.
  • Prime factorization: 1440 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5, which can be written 1440 = 2⁵ × 3² × 5
  • 1440 has at least one exponent greater than 1 in its prime factorization so √1440 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1440 = (√144)(√10) = 12√10
  • The exponents in the prime factorization are 5,2, and 1. Adding one to each exponent and multiplying we get (5 + 1)(2 + 1)(1 + 1) = 6 × 3 × 2 = 36. Therefore 1440 has exactly 36 factors.
  • The factors of 1440 are outlined with their factor pair partners in the graphic below.

There you have it. The exponents of the prime factorization determine the number of factors a number has, and (5 + 1)(2 + 1)(1 + 1) = 6 × 3 × 2 = 36. How do we find its prime factorization? We can divide it over and over by the smallest prime number factor until we make an easy-to-read and very delicious cake!

Or you could make a factor tree such as the one below. Then rake up the prime number factors afterward and hope that you see them all:

I prefer cake to yard work any day!

As I mentioned before, 1440 is NOT the smallest number with 36 factors:
Because (2 + 1)(2 + 1)(1 + 1)(1 + 1) = 3 × 3 × 2 × 2 = 36 also, we get
2²× 3²× 5 × 7 = 1260, which just happens to be smaller than 1440, so 1260 gets the smallest-number-with-36-factors prize. 1440 is the SECOND smallest number with 36 factors.

1440 is the sum of the interior angles of a decagon. Why?
Because 180(10-2) = 180(8) = 1440
So what? 1260 is the sum of the interior angles of a nonagon.

1440 is the hypotenuse of only one Pythagorean triple:
864-1152-1440 which is (3-4-5) times 288
Yeah? 1260 is the hypotenuse of only one Pythagorean triple as well.

Look, 1440 is not second best because it has this one other claim to fame:
From Stetson.edu we learn that 1440 deserves a lot of exclamation points since
1440 = 2!3!5!

There you have it. We’ve taken the mystery out of the number 1440, AND it is a fabulous number!

 

1431 is a Triangular Number and a Hexagonal Number

If you only look at a list of triangular numbers or a list of hexagonal numbers, you might miss the relationship that figurate numbers have with each other.

1378 is the 52nd triangular number, and you can use it to find the 53rd triangular number (1431), the 53rd square number, the 53rd pentagonal number, and so forth.

351 is the 26th triangular number, and you can use it to find the 27th triangular number, the 27th square number, the 27th pentagonal number, the 27th hexagonal number (1431), and so forth.

See the relationship in the graphic below:

Should you get excited that 1431 is BOTH a triangular number and a hexagonal number? Not really. It turns out that every hexagonal number is also a triangular number. (But not every triangular number is a hexagonal number.)

Here are a few more facts about the number 1431:

  • 1431 is a composite number.
  • Prime factorization: 1431 = 3 × 3 × 3 × 53, which can be written 1431 = 3³ × 53
  • 1431 has at least one exponent greater than 1 in its prime factorization so √1431 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1431 = (√9)(√159) = 3√159
  • 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 1431 has exactly 8 factors.
  • The factors of 1431 are outlined with their factor pair partners in the graphic below.

1431 is the hypotenuse of a Pythagorean triple:
756-1215-1431 which is 27 times (28-45-53)