Mathematics

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:

Finding Primes Like 1367

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Ancient Greek Eratosthenes had a method of finding prime numbers. We call it the Sieve of Eratosthenes. You can use this method if you make a list of numbers, circle the first available prime number, cross out all of its multiples on the chart and repeat tat procedure over and over again. The next number that has not been crossed out is the next prime number. Numbers that get crossed out are composite numbers, and they can always be expressed as the product of prime numbers.

Many teachers have given their students a 100 chart to help them find the twenty-five prime numbers less than 100.

If the number 1 were a prime number, we would have to cross out all of its multiples, and that would make 1 be the only prime number!?? That would be an unacceptable conclusion. It turns out that 1 cannot be either prime or composite. Perhaps you will want to put a star around it.

Here is a 100 chart with the multiples of 2, 3, 5, and 7 crossed out.

Since 10 is a multiple of 2 and 5, it was easy to cross out their multiples. 9, a multiple of 3, is one less than 10, so 3’s multiples were also easy to cross out. Crossing out all the multiples of 7 is a little bit tedious at first, but you can even find a pattern for them as well.

Notice that the distance between multiples of any given prime is that prime number whether you count horizontally or vertically. The circled numbers AND all the numbers not crossed out on the chart are prime numbers.

This Sieve of Eratosthenes is a powerful method for finding prime numbers, but some of its power is lost when just a 100 chart is used. For pretty much the same amount of work, we could have used the list of numbers in a 10 × 12 chart because 10 × 12 = 120 which is one less than the next prime number squared, 11² = 121.

But wait a minute! 11 is one more than 10 so its multiples would be SO easy to cross out. Let’s make the chart go to one less than 168 which is one less than 13²:

Yeah, it’s not a perfect rectangle anymore, but for about the same effort, we get an additional fourteen prime numbers. I also like that all of these multiples of seven {21, 42, 63, 84, 105, 126, 147, 168} are sort of on a diagonal. What do you notice about the numbers in that set? The next number in the set isn’t on the chart, but can you figure out what it is and where it goes?

To find out if a number like 1367 is prime, I wouldn’t want to expand a 100 chart into a 1400 chart. That chart would also require me to cross out all the multiples of every prime number from 13 to 31, and that might be a big pain.  It’s much easier to see if 1367 is divisible by any of the prime numbers less than its square root. It isn’t, so I conclude:

  • 1367 is a prime number.
  • Prime factorization: 1367 is prime.
  • 1367 has no exponents greater than 1 in its prime factorization, so √1367 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1367 has exactly 2 factors.
  • The factors of 1367 are outlined with their factor pair partners in the graphic below.

How do we know that 1367 is a prime number? If 1367 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1367. Since 1367 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1367 is a prime number.

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?

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.

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

1348 Coloring Paula Krieg’s Polar Rose

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Paula Beardell Krieg recently wrote about using Desmos to create designs that can be colored by hand or by computer programs like Paint. I like using Paint so with her permission I took a design she made and colored it so I could present it here in this post. I chose colors that make me think of spring because, frankly, I’m ready for winter to be over!

Now I’ll write a little bit about the number 1348:

  • 1348 is a composite number.
  • Prime factorization: 1348 = 2 × 2 × 337, which can be written 1348 = 2² × 337
  • 1348 has at least one exponent greater than 1 in its prime factorization so √1348 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1348 = (√4)(√337) = 2√337
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1348 has exactly 6 factors.
  • The factors of 1348 are outlined with their factor pair partners in the graphic below.

1348 is the sum of two squares:
32² + 18² = 1348

1348 is the hypotenuse of a Pythagorean triple:
700-1152-1348 which is 32² – 18², 2(32)(18), 32² + 18²

1348 is also the short leg in a primitive Pythagorean triple:
1348-454275-454277

 

 

1347 and Level 1

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Which ten division facts do you need to know to solve this puzzle? Seriously, you can do this one!

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you’re not sure how to solve it, I explain how in this youtube video:

Here are some facts about the number 1347:

  • 1347 is a composite number.
  • Prime factorization: 1347 = 3 × 449
  • 1347 has no exponents greater than 1 in its prime factorization, so √1347 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 1347 has exactly 4 factors.
  • The factors of 1347 are outlined with their factor pair partners in the graphic below.

1347 isn’t a Lucas number, but as OEIS.org reminds us, 1, 3, 4, 7 happens to be the first four Lucas numbers. Here’s why:
Start with 1, 3, then . . .
1 + 3 = 4
3 + 4 = 7
So what will be the next Lucas number?

1347 is the hypotenuse of a Pythagorean triple:
840-1053-1347 which is 3 times (280-351-449)

(1346÷2)×3 = 2019 Magic or Square?

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I was inspired to make a 3 × 3 Magic Square where every number is different but the numbers in each row, column and diagonal added together equal the same number, 54:

I made it by taking a regular 3 × 3 Magic Square and adding 13 to each of its numbers.

What inspired me to do that? This magical tweet of a palindromic Magic Square for the Year 2019:

Yeah, I know my magic square isn’t quite as impressive. It might be more square than it is magic. It’s also less impressive than this number 2019 spelled out using fifty-one
4 × 4 Magic Squares.

Maybe you will be more impressed by this magic square that has 2019 as its Magic Sum?

I could make that magic sum because 2019 is divisible by 3. Why is 673 in the center? Because 2019÷3 = 673.

673 × 2 = 1346. I’m sharing these magic squares in this post I’ve numbered 1346. In case you haven’t figured it out (1346÷2)×3 = 2019. Happy New Year!

Here’s more about the number 1346:

  • 1346 is a composite number.
  • Prime factorization: 1346 = 2 × 673
  • 1346 has no exponents greater than 1 in its prime factorization, so √1346 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 1346 has exactly 4 factors.
  • The factors of 1346 are outlined with their factor pair partners in the graphic below.

1346 is the hypotenuse of a Pythagorean triple:
770-1104-1346 which is 2 times 385-552-673

How Far Away Is 1344 from the Nearest Prime Number?

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The Distance 1344 is from the Nearest Prime Number:

1344 is 17 numbers away from the nearest prime number. 17-away is a new record for distance to the closest prime number!

1344 will hold that record until prime number 2179 claims it with being 18-away from the nearest prime:
2161–(18 composite numbers)–2179–(24 composite numbers)–2203

OEIS.org mentioned the previous two numbers, 1342 and 1343, breaking the 15-away and the 16 away records respectively, but didn’t mention this one. I guess it considered 1344 being the order of a perfect group to be more important.

Factors of 1344:

  • 1344 is a composite number.
  • Prime factorization: 1344 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 7, which can be written 1344 = 2⁶ × 3 × 7
  • 1344 has at least one exponent greater than 1 in its prime factorization so √1344 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1344 = (√64)(√21) = 8√21
  • The exponents in the prime factorization are 6, 1 and 1. Adding one to each exponent and multiplying we get (6 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 1344 has exactly 28 factors.
  • The factors of 1344 are outlined with their factor pairs in the graphic below.

Sum-Difference Puzzles:

84 has six factor pairs. One of those pairs adds up to 25, and another one subtracts to 25. Put the factors in the appropriate boxes in the first puzzle.

1344 has fourteen factor pairs. One of the factor pairs adds up to ­100, and a different one subtracts to 100. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?