Prime factorization

1376 Let’s Get Ready for the Playful Math Carnival!

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Too many people think that mathematics is a house of horrors, but there are plenty of bloggers out there, who know that done right, math is actually ALL fun and games. It is like a carnival! Every month, you can play at the Playful Math Education Blog Carnival, and it really is play! What does a playful math carnival look like? Go on over to see how Math Mama Writes… and puts on a fabulous March carnival!

I will be hosting this monthly carnival the last week of April! Why do I get to host it? I sent a message on twitter to Denise Gaskins who coordinates the carnival, and I requested the privilege. If you would like to host it in the future, let her know. She is always looking for blogs to host, and she will be very happy to hear from you.

In the meantime, you can help me with my carnival. If you blog about mathematics in a playful way that could benefit children who are somewhere between preschool to high school age, I would love to include your post in my carnival. The carnival is a FREE way to promote your post, so if you would like more traffic to your blog, submit a post using the link from Denise Gaskins’ website by Friday, April 19. Then before the end of the month, you will be able to enjoy the carnival even more because of your participation!

Now I’ll tell you a little bit about the post number, 1376:

  • 1376 is a composite number.
  • Prime factorization: 1376 = 2 × 2 × 2 × 2 × 2 × 43, which can be written 1376 = 2⁵ × 43
  • 1376 has at least one exponent greater than 1 in its prime factorization so √1376 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1376 = (√16)(√86) = 4√86
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1376 has exactly 12 factors.
  • The factors of 1376 are outlined with their factor pair partners in the graphic below.

As mentioned in my previous post, 1376 is part of the three smallest consecutive numbers that have cube roots that can be simplified.

The Cube Root of 1375 is the Smallest. . .

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The cube root of 1375 can be simplified. So can the cube roots of 1376 and 1377. There are no smaller consecutive three numbers whose cube roots can make the same claim.

Thank you, OEIS.org for alerting me to that very cool fact. It deserves a celebration so I made the graphic above.

Here’s more about the number 1375:

  • 1375 is a composite number.
  • Prime factorization: 1375 = 5 × 5 × 5 × 11, which can be written 1375 = 5³ × 11
  • 1375 has at least one exponent greater than 1 in its prime factorization so √1375 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1375 = (√25)(√55) = 5√55
  • 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 1375 has exactly 8 factors.
  • The factors of 1375 are outlined with their factor pair partners in the graphic below.

1375 is also the hypotenuse of THREE Pythagorean triples:
385-1320-1375 which is (7-24-25) times 55
484-1287-1375 which is 11 times (44-117-125)
825-1100-1375 which is (3-4-5) times 275

 

 

1374 and Level 2

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Can you find the factors that will turn this puzzle into a multiplication table whose products are simply not in the usual order?

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are a few facts about the puzzle number, 1374:

  • 1374 is a composite number.
  • Prime factorization: 1374 = 2 × 3 × 229
  • 1374 has no exponents greater than 1 in its prime factorization, so √1374 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 1374 has exactly 8 factors.
  • The factors of 1374 are outlined with their factor pair partners in the graphic below.

1374 is the hypotenuse of a Pythagorean triple:
360-1326-1374 which is 6 times (60-221-229)

An Efficient Way to Quickly Find All the Odd Prime Numbers Less Than 1369

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As I’ve used different sized grids to play with the Sieve of Eratosthenes, I’ve decided I like one size grid better than all the rest. It has six odd numbers across, but I repeat the first column on the right because of convenience. You already know the only even prime number is 2, so this grid can help you find all the rest of the primes up to 1369 = 37².

Look at the grid. What are some things that you notice about it?

Square numbers are never prime, so why do I have them outlined on the grid? Why are some of them crossed out? Is there a pattern for that, too?

If you’ve done a sieve where you cross out all the multiples of the prime numbers in order, perhaps you’ve noticed that the first multiple to get crossed out that hasn’t been crossed out before is always the prime number squared.

Therefore, don’t start with the prime numbers. Start with their squares! The squares of each of the prime numbers and the next five odd multiples after those squares are listed in a box on the left of the paper. Put a dot in the corner of each of those multiples. Recognize the pattern they make and strike through those numbers with a colored pencil. A ruler will be helpful. Continue the same pattern down to the bottom of the grid. Then do the same thing with the next square of a prime number.  I’ve made a gif of these instructions being applied to a much smaller grid.
Finding Primes Less Than 361

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It feels like I’m wrapping twinkling lights around a long sheet of cardboard!

Do try it on this much longer grid that goes to 1369. You’ll probably want to cut it out and glue or tape it together on the back. If this is an assignment, don’t cut off your name.

Read the following AFTER you’ve tried using the grid. I don’t want to spoil your sense of discovery!

To me, the lines drawn have a slope even if the lines are broken lines.
The slope for the 3s is undefined.
For the 5s, it’s +1; for the 7’s, it’s -1;
For the 11’s, it’s +2; for the 13’s, it’s -2;
For the 17’s, it’s +3; for the 19’s, it’s -3;
For the 23’s, it’s +4; for the 25’s, it’s -4; (You can skip the 25’s because they are already crossed out.)
For the 29’s, it’s +5; for the 31’s, it’s -5;
Cross out 37², and then you are done.

If the grid were longer, you could continue with the same pattern for as long as you want. I think it is pretty cool.

Now I’ll tell you a little bit about the number 1369:

  • 1369 is a composite number and is a perfect square.
  • Prime factorization: 1369 = 37 × 37 which can be written 1369 = 37²
  • 1369 has at least one exponent greater than 1 in its prime factorization so √1369 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1369 = (√37)(√37) = 37
  • The exponent in the prime factorization is 2. Adding one to that exponent, we get (2 + 1) = 3. Therefore 1369 has exactly 3 factors.
  • The factors of 1369 are outlined with their factor pair partners in the graphic below.

1369 is the sum of two squares:
35² + 12² = 1369

1369 is the hypotenuse of two Pythagorean triples:
444-1295-1369 which is (12-35-37) times 37
840-1081-1369 calculated from 2(35)(12), 35² – 12², 35² + 12²

 

1368 Playing with the Sieve of Eratosthenes

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What if we didn’t use ten numbers across for our Sieve of Eratosthenes?

For example, 36 × 38 = 1368. We could make a Sieve of Eratosthenes writing 36 numbers across the grid and 38 numbers down. The last number on the grid would be 1368, and we could find all the prime numbers less than 1369 (which is 17²) by crossing out all the multiples of the prime numbers that appear on the top row. The trouble is that 36 numbers across makes a very big grid. Crossing out multiples of 2, 3, 5, and 7 will be very quick, but crossing out all the multiples of 11, 13, 17, 19, 23, 29, and 31 will not be so fun.

Grids that make use of the fact that (n-1)(n+1) = n²-1 can always give us a perfect rectangle and we will only need to cross out the multiples of the prime numbers in the top row to find ALL the prime numbers in any (n-1)×(n+1) list of numbers.

Here’s a 7 × 9 grid:

Since it was 7 across, it was very easy to cross out all the multiples of 7. The multiples of 2 and 3 weren’t too difficult to find either, but the pattern for the multiples of 5 was not quite as nice. Fortunately, it is easy to spot those multiples, no matter how big a number they are.

Still, the first prime number on the second row is 11, so we should be able to go almost up to 11² = 121 on our grid:

I couldn’t fit 120 on the grid without ruining the rectangle, but here’s a grid using 12 numbers across. Since 12 × 14 = 168 which is one less than 13², we can find all the prime numbers in the list simply by crossing out the multiples of the prime numbers in the top row.

But the next number, 13, is only one number more than 12, and all of its multiples are staring at me making me feel very uncomfortable. It will be very easy to cross out all of the multiples of 13. That means we can extend the list of numbers to one less than the next prime number squared, which is 289 – 1 = 288. This time we get a perfect rectangle because 288 is also a multiple of 12:

All of the circled numbers in the top row and every number that has not been crossed out below the top row are prime numbers.

Someone long ago figured out that if we make the grid six numbers across, all the prime numbers except 2 and 3 will appear in the same two columns, no matter how long the grid is:

Every prime number greater than 3 is either one less or one more than a multiple of 6.

Since we always cross out the multiples of 2 anyway, what would happen if we didn’t include them in the grid at all?

Here is a grid with ten numbers across, but only odd numbers are included. Because 5 is a factor of 10, it is very easy to cross out all of the 5’s. Also, since 9 is one less than 10 and 11 is one more than 10, it is also easy to cross out all the multiples of 3 and 11. Crossing out the 7’s and the 19’s wasn’t too bad, either, but the 13’s and 17’s were not as fun.

In my next post, I will share my favorite size of grid and the method I use to find all of the prime numbers on it. No prime numbers get circled in my method.

Some of the numbers in the grids had several lines through them.
If we made the 36 × 38 grid I mentioned at the beginning of the post, how many lines would 1368 have going through it?  After all, 1368 has 24 factors. What do you think?

Only three lines. One each for its prime factors, 2, 3 and 19.

Here’s more about the number 1368:

  • 1368 is a composite number.
  • Prime factorization: 1368 = 2 × 2 × 2 × 3 × 3 × 19, which can be written 1368 = 2³ × 3² × 19
  • 1368 has at least one exponent greater than 1 in its prime factorization so √1368 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1368 = (√36)(√38) = 6√38
  • The exponents in the prime factorization are 3, 2, and 1. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1368 has exactly 24 factors.
  • The factors of 1368 are outlined with their factor pair partners in the graphic below.

Here’s one of the MANY possible factor trees for 1368:

1366 Fractions Acting Improperly

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In elementary school, we learned about improper fractions. Should we call them that? Is it even possible for any kind of number to be IMPROPER? They are simply fractions greater than one. I’ve recently heard the term fraction form used, and ever since I’ve made a point of saying that fractions greater than one are in fraction form.

On Twitter, I’ve found a few people who also don’t like using the word improper to describe any fraction.

This first tweet has a link explaining why it is improper to use the term improper fraction:

I hope that you will consider not labeling any fraction as improper, as well!

Now I’ll write a little bit about the number 1366:

  • 1366 is a composite number.
  • Prime factorization: 1366 = 2 × 683
  • 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 1366 has exactly 4 factors.
  • Factors of 1366: 1, 2, 683, 1366
  • Factor pairs: 1366 = 1 × 1366 or 2 × 683
  • 1366 has no square factors that allow its square root to be simplified. √1366 ≈ 36.95944

1366 is also the sum of the twenty-six prime number from 5 to 107. Do you know what all those prime numbers are?

1365 Shamrock Mystery

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Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story.  Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you!

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Now I’ll share some information about the number 1365:

  • 1365 is a composite number.
  • Prime factorization: 1365 = 3 × 5 × 7 × 13
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1365 has exactly 16 factors.
  • Factors of 1365: 1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365
  • Factor pairs: 1365 = 1 × 1365, 3 × 455, 5 × 273, 7 × 195, 13 × 105, 15 × 91, 21 × 65, or 35 × 39
  • 1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591

1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273

1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16

I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help:

1364 is the 15th Lucas Number

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OEIS.org reminded me that the 15th Lucas Number is 1364, so I’ve made a graphic illustrating that fact:

In this 40-second video I explain how to generate that list in excel using the drag feature:

Here are a few more facts about the number 1364:

  • 1364 is a composite number.
  • Prime factorization: 1364 = 2 × 2 × 11 × 31, which can be written 1364 = 2² × 11 × 31
  • 1364 has at least one exponent greater than 1 in its prime factorization so √1364 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1364 = (√4)(√341) = 2√341
  • 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 1364 has exactly 12 factors.
  • The factors of 1364 are outlined with their factor pair partners in the graphic below.

1363 and Level 6

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The common factors of 60 and 30 allowed in the puzzle are 5, 6, and 10. Which one is the logical choice? Look at the other clues in the puzzle and you should be able to eliminate two of the choices.

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few thoughts about the puzzle number, 1363:

  • 1363 is a composite number.
  • Prime factorization: 1363 = 29 × 47
  • 1363 has no exponents greater than 1 in its prime factorization, so √1363 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 1363 has exactly 4 factors.
  • The factors of 1363 are outlined with their factor pair partners in the graphic below.

1363 is the hypotenuse of a Pythagorean triple:
940-987-1363 which is (20-21-29) times 47

What Is Special about √1362

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What is special about √1362? I don’t mean to sound like a Chevy commercial, but Just Look At It! I made this gif to show off √1362:

Square Root 1362

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1362 is the smallest number whose square root does that. Yeah, there have been some numbers that came close, but this one includes zero! Thank you OEIS.org for alerting me to that fact.
Here are a few more facts about the number 1362:
  • 1362 is a composite number.
  • Prime factorization: 1362 = 2 × 3 × 227
  • 1362 has no exponents greater than 1 in its prime factorization, so √1362 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 1362 has exactly 8 factors.
  • The factors of 1362 are outlined with their factor pair partners in the graphic below.
You can easily calculate a couple of monstrous Pythagorean triples that contain 1362:
1362-463760-463762 calculated from 2(681)(1), 681² – 1²,  681² + 1², and
1362-51520-51538 calculated from 2(227)(3), 227² – 3²,  227² + 3²