1483 and Level 5

Today’s Puzzle:

Some of the clues in the same row or column in this puzzle have more than one common factor. In each case, will you make the logical choice to find the puzzle’s unique solution?

Factors of 1483:

  • 1483 is a prime number.
  • Prime factorization: 1483 is prime.
  • 1483 has no exponents greater than 1 in its prime factorization, so √1483 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1483 has exactly 2 factors.
  • The factors of 1483 are outlined with their factor pair partners in the graphic below.

How do we know that 1483 is a prime number? If 1483 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1483. Since 1483 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1483 is a prime number.

More about the Number 1483:

1483 is the difference of two squares:
742² – 741² = 1483

The first five prime decades are listed below. 1483 is the second prime number in the fifth prime decade:

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1481 is the First Prime Number in the Fifth Prime Decade

What Kind of Prime Number is 1481?

1481 is the 234th prime number. It is part of a twin prime pair, a prime triplet, a prime quadruplet, and even a prime quintuplet. All of those designations are nice, but to me, the most beautiful is the prime decade (the prime quadruplets not counting (5, 7, 11, 13)).

1481 is the first prime number in the fifth prime decade. The last digits of the numbers in every prime decade make a lovely pattern: 1, 3, 7, and 9. Once you know the first prime number in the decade, you also know the other three! That’s beautiful!

I found the numbers in the first five prime decades easy to memorize. We start with the teen decade followed by the first and last decades in the 100’s. Then199 is almost 200 and that helps me remember 821, and 821 helps me remember 1481.

Factors of 1481:

  • 1481 is a prime number.
  • Prime factorization: 1481 is prime.
  • 1481 has no exponents greater than 1 in its prime factorization, so √1481 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1481 has exactly 2 factors.
  • The factors of 1481 are outlined with their factor pair partners in the graphic below.

How do we know that 1481 is a prime number? If 1481 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1481. Since 1481 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1481 is a prime number.

More about the Number 1481:

1481 is the sum of two squares:
35² + 16² = 1481.

1481 is the hypotenuse of a Pythagorean triple:
969-1120-1481, calculated from 35² – 16², 2(35)(16), 35² + 16².

Here’s another way we know that 1481 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 16² = 1481 with 35 and 16 having no common prime factors, 1481 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1481. Since 1481 is not divisible by 5, 13, 17, 29, or 37, we know that 1481 is a prime number.

 

1471 and Level 3

Today’s Puzzle:

There is only one factor that will work with the clue in the top row of this level 3 puzzle. Find it, then work your way down row by row until you have found all the factors. There is only one solution.

Factors of 147

  • 1471 is a prime number.
  • Prime factorization: 1471 is prime.
  • 1471 has no exponents greater than 1 in its prime factorization, so √1471 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1471 has exactly 2 factors.
  • The factors of 1471 are outlined with their factor pair partners in the graphic below.

How do we know that 1471 is a prime number? If 1471 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1471. Since 1471 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1471 is a prime number.

Another Fact about the Number 1471:

1471 is the 21st centered heptagonal number because
(7(21²)-7(21)+2)/2 = 1471.

1459 and Level 4

Today’s Puzzle:

Can you place the numbers 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues? You might surprise yourself with how well you do!

Factors of 1459:

  • 1459 is a prime number.
  • Prime factorization: 1459 is prime.
  • 1459 has no exponents greater than 1 in its prime factorization, so √1459 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1459 has exactly 2 factors.
  • The factors of 1459 are outlined with their factor pair partners in the graphic below.

How do we know that 1459 is a prime number? If 1459 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1459. Since 1459 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1459 is a prime number.

Other Facts about the number 1459:

1459 is the difference of two squares:
730² – 729² = 1459

From Stetson.edu we learn this chilling cubic fact:
1³ + 4³ + 5³ + 9³ = 919, and
9³ + 1³ + 9³ = 1459

 

1453 Happy Birthday, Jo Morgan

Who Is Jo Morgan?

Jo Morgan is an inspiring mathematics teacher, collaborator, tweeter, blogger, podcast guest, and author. Today is her birthday!

Jo recently published her first book, A Compendium of Mathematical Methodsand it is all the rage on twitter. Here is a small sampling of tweets expressing excitement for her book:

I can hardly wait until February 4th when Amazon makes it available in the United States!

Jo has enjoyed solving some of my puzzles, so to commemorate her birthday, I’ve made one especially for her. To solve this puzzle, write the numbers 1 to 10 in each of the four sections outlined in bold so that those numbers are the factors of the product clues given in each of the four mixed-up multiplication tables that make up the puzzle. Use logic to solve the puzzle, but I’m warning you, it won’t be easy.

Happy birthday, Jo! I hope you enjoy the puzzle!

Find the Factors 1 – 10 Birthday Challenge Puzzle:

Print the puzzles or type the solution in this excel file: 12 Factors 1443-1453

Factors of 1453:

It is convenient for puzzles to be numbered, and this puzzle number is 1453. Here are a few facts about that number:

  • 1453 is a prime number.
  • Prime factorization: 1453 is prime.
  • 1453 has no exponents greater than 1 in its prime factorization, so √1453 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1453 has exactly 2 factors.
  • The factors of 1453 are outlined with their factor pair partners in the graphic below.

How do we know that 1453 is a prime number? If 1453 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1453. Since 1453 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1453 is a prime number.

Other Facts about the Number 1453:

1453 is the sum of two squares:
38² + 3² = 1453

That means 1453 is the hypotenuse of a Pythagorean triple:
228-1435-1453 calculated from 2(38)(3), 38² – 3², 38² + 3².

Here’s another way we know that 1453 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 3² = 1453 with 38 and 3 having no common prime factors, 1453 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1453. Since 1453 is not divisible by 5, 13, 17, 29, or 37, we know that 1453 is a prime number.

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.

1439 and Level 6

You know the factors of 49, but to complete this puzzle, you will also have to determine the answers to a few questions:

Will the common factor of 24 and 32 be 4 or 8?
Will the common factor of 20 and 40 be 4, 5, or 10?
Will the common factor of 27 and 9 be 3 or 9?

Don’t guess and check! Look at the other clues, and some of the possibilities will be eliminated*. Once you’ve found the next clue to use, continue using logic until the whole puzzle is completed.

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

  • 1439 is a prime number.
  • Prime factorization: 1439 is prime.
  • 1439 has no exponents greater than 1 in its prime factorization, so √1439 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1439 has exactly 2 factors.
  • The factors of 1439 are outlined with their factor pair partners in the graphic below.

How do we know that 1439 is a prime number? If 1439 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1439. Since 1439 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1439 is a prime number.

* Did you notice that either one of the 24’s or 6 must use a 3 because only two 6s are allowed to be used as factors? What does that tell you about the common factor of 27 and 9?

 

1429 Find the Factors Challenge

I wanted today’s puzzle to look like a big candy bar, but I don’t think I succeeded. I hope you will still think it is the best treat you got today! Good luck!

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you some facts about the number 1429:

  • 1429 is a prime number.
  • Prime factorization: 1429 is prime.
  • 1429 has no exponents greater than 1 in its prime factorization, so √1429 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1429 has exactly 2 factors.
  • The factors of 1429 are outlined with their factor pair partners in the graphic below.

How do we know that 1429 is a prime number? If 1429 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1429. Since 1429 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1429 is a prime number.

Stetson.edu informs us that 1429² = 2,042,041. That’s the smallest perfect square whose first three digits are repeated in order by the next three digits.

1429 is the sum of two squares:
30² + 23² = 1429

1429 is the hypotenuse of a primitive Pythagorean triple:
371-1380-1429 calculated from 30² – 23², 2(30)(23), 30² + 23²

Here’s another way we know that 1429 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 23² = 1429 with 30 and 23 having no common prime factors, 1429 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1429. Since 1429 is not divisible by 5, 13, 17, 29, or 37, we know that 1429 is a prime number.

 

1409’s Super Power

 

Stetson.edu informed me that 1409⁸ is the ONLY known 8th power that is the sum of EIGHT 8th powers. Wow! That seems to me to give 1409 quite the superpower!

What were those eight 8th powers that are included in the sum? That’s a puzzle more suited for a computer than a human, but Wolfram Mathworld Diophantine came to my rescue with this POWERFUL fact: 1324⁸+1190⁸+1088⁸+748⁸+524⁸+478⁸+223⁸+90⁸=1409⁸.

Go ahead and check it out on your computer’s calculator. It’s true! Notice also that two of those eighth powers are permutations of each other!

I was so intrigued with 1409 that I had to make this cape so everyone can see how super 1409 is:

Sometimes 1409 wears a more modest super cape because 1409² is also the sum of TWO squares:
159² +1400² = 1409² 

Here are some more super facts about the number 1409:

  • 1409 is a prime number.
  • Prime factorization: 1409 is prime.
  • 1409 has no exponents greater than 1 in its prime factorization, so √1409 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1409 has exactly 2 factors.
  • The factors of 1409 are outlined with their factor pair partners in the graphic below.

How do we know that 1409 is a prime number? If 1409 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1409. Since 1409 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1409 is a prime number.

1409 is the sum of two squares:
28² + 25² = 1409

1409 is the hypotenuse of a primitive Pythagorean triple:
159-1400-1409 calculated from 28² – 25², 2(28)(25), 28² + 25²

Here’s another way we know that 1409 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 25² = 1409 with 28 and 25 having no common prime factors, 1409 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1409. Since 1409 is not divisible by 5, 13, 17, 29, or 37, we know that 1409 is a prime number.

 

1399 and Level 6

The eligible common factors of 48 and 72 are 6, 8, and 12. The common factors for 10 and 30 are 5 and 10.  Don’t guess and check the possibilities! Can you figure out the logic needed to start this puzzle?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s a little information about the number 1399:

  • 1399 is a prime number.
  • Prime factorization: 1399 is prime.
  • 1399 has no exponents greater than 1 in its prime factorization, so √1399 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1399 has exactly 2 factors.
  • The factors of 1399 are outlined with their factor pair partners in the graphic below.

How do we know that 1399 is a prime number? If 1399 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1399. Since 1399 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1399 is a prime number.

1399 is the difference of two squares:
700² – 699² = 1399