1368 Playing with the Sieve of Eratosthenes

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:

Advertisements

1328 Christmas Tree

 

Can you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that the lights on this Christmas tree work properly?

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Now I’ll share some facts about the puzzle number, 1328:

  • 1328 is a composite number.
  • Prime factorization: 1328 = 2 × 2 × 2 × 2 × 83, which can be written 1328 = 2⁴ × 83
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1328 has exactly 10 factors.
  • Factors of 1328: 1, 2, 4, 8, 16, 83, 166, 332, 664, 1328
  • Factor pairs: 1328 = 1 × 1328, 2 × 664, 4 × 332, 8 × 166, or 16 × 83
  • Taking the factor pair with the largest square number factor, we get √1328 = (√16)(√83) = 4√83 ≈ 36.44173

 

Because 28 is divisible by 4, but not by 8, and 3 (the digit before the 28) is an odd number, I know that 1328 is divisible by 8. I can use that fact to make this simple factor tree:

1328 is the difference of two squares three different ways:
333² – 331² = 1328
168² – 164² = 1328
87² – 79²  = 1328

1248 Factor Trees

It is easy to see that 1248 is divisible by 12 and therefore by 2, 3, 4, and 6, but it actually has a lot more factors than that. I decided to celebrate its many factors by making just a few of its many possible factor trees.

Why does 1248 have so many factors? Well, check out its prime factorization:

  • 1248 is a composite number.
  • Prime factorization: 1248 = 2 × 2 × 2 × 2 × 2 × 3 × 13, which can be written 1248 = 2⁵ × 3 × 13
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1248 has exactly 24 factors.
  • Factors of 1248: 1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 32, 39, 48, 52, 78, 96, 104, 156, 208, 312, 416, 624, 1248
  • Factor pairs: 1248 = 1 × 1248, 2 × 624, 3 × 416, 4 × 312, 6 × 208, 8 × 156, 12 × 104, 13 × 96, 16 × 78, 24 × 52, 26 × 48, or 32 × 39
  • Taking the factor pair with the largest square number factor, we get √1248 = (√16)(√78) = 4√78 ≈ 35.32704

1248 is the sum of four consecutive prime numbers:
307 + 311 + 313 + 317 = 1248

1248 is also the hypotenuse of a Pythagorean triple:
480-1152-1248 which is (5-12-13) times 96

How I Knew Immediately that a Factor Pair of 1224 is . . .

12 = 3 × 4 and 24 is one less than 25. Those two facts helped me to know right away that 35² = 1225 and 34 × 36 = 1224. Study the patterns in the chart below and you will likely be able to remember all of the multiplication facts listed in it!

a² – b² = (a – b)(a + b)
You may remember how to factor that from algebra class. Here when b = 1, it has a practical application that can allow you to amaze your friends and family with your mental calculating abilities!

I’ve only typed a small part of that infinite pattern chart. For example, if you know that 19 × 20 = 380, then you can also know that 195² = 38025 and 194 × 196 = 38024.

Also because of that chart, I know that 3.5² = 12.25 and 3.4 × 3.6 = 12.24
(Also (3½)² = 12¼, but 2½  × 4½ = 11¼ because 3-1 = 2, 3+1 = 4, 12-1 = 11
thus 2.5 × 4.5 = 11.25 and 2½  × 4½ = 11¼)

You could also let b = 2 so b² = 4. Then 25 – 4 = 21, and you could know facts like
33 × 37 = 1221 or 193 ×  197 = 38021

I hope you have a wonderful time being a calculating genius!

Now I’ll share some other facts about the number 1224:

  • 1224 is a composite number.
  • Prime factorization: 1224 = 2 × 2 × 2 × 3 × 3 × 17, which can be written 1224 = 2³ × 3² × 17
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1224 has exactly 24 factors.
  • Factors of 1224: 1, 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72, 102, 136, 153, 204, 306, 408, 612, 1224
  • Factor pairs: 1224 = 1 × 1224, 2 × 612, 3 × 408, 4 × 306, 6 × 204, 8 × 153, 9 × 136, 12 × 102, 17 × 72, 18 × 68, 24 × 51 or 34 × 36
  • Taking the factor pair with the largest square number factor, we get √1224 = (√36)(√34) = 6√34 ≈ 34.98571

When a number has so many factors, I often will make a forest of factor trees for that number, but today I just want us to enjoy this one tree for 34 × 36 = 1224.

1224 is also the sum of two squares:
30² + 18² = 1224

1224 is the hypotenuse of a Pythagorean triple:
576-1080-1224 which is (8-15-17) times 72
That triple can also be calculated from 30² – 18², 2(30)(18), 30² + 18²

293 + 307 + 311 + 313 = 1224 making 1224 the sum of four consecutive prime numbers.

1170 Factor Trees

1170 is one of those numbers with twenty-four factors. Why does it have so many? Because of its prime factorization. You can find a number’s prime factorization by making a factor tree. Here are eleven different factor trees for 1170. Each one of them produces the same prime factorization: 1170 = 2 × 3² × 5 × 13

  • 1170 is a composite number.
  • Prime factorization: 1170 = 2 × 3 × 3 × 5 × 13, which can be written 1170 = 2 × 3² × 5 × 13
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1170 has exactly 24 factors.
  • Factors of 1170: 1, 2, 3, 5, 6, 9, 10, 13, 15, 18, 26, 30, 39, 45, 65, 78, 90, 117, 130, 195, 234, 390, 585, 1170
  • Factor pairs: 1170 = 1 × 1170, 2 × 585, 3 × 390, 5 × 234, 6 × 195, 9 × 130, 10 × 117, 13 × 90, 15 × 78, 18 × 65, 26 × 45, or 30 × 39
  • Taking the factor pair with the largest square number factor, we get √1170 = (√9)(√130) = 3√130 ≈ 34.20526

33² +  9² = 1170
27² +  21² = 1170

1170 is also the hypotenuse of FOUR Pythagorean triples:
288-1134-1170 calculated from 27² –  21², 2(27)(21), 27² +  21²
450-1080-1170 which is (5-12-13) times 90
594-1008-1170 calculated from 2(33)( 9), 33² –  9², 33² +  9²
702-936-1170 which is (3-4-5) times 234

1170 is 102102 in BASE 4 because 4⁵ + 2(4³) +4² + 2(1) = 1170,
and it’s repdigit 2222 in BASE 8 because 2(8³ + 8² + 8¹ + 8⁰) = 2(585) = 1170

1152 Will You See All the Prime Factors in This Factor Tree?

Will you see all the prime factors in this factor tree when all of the factors are the same color? You know how to count, but would you possibly not count one of the prime factors or possibly count one of them twice?

Is it easier to count the prime factors in the following factor trees?

If you happen to have two different colors of ink and/or pencils around when you make factor trees, they might be easier to read especially if the factored number has a lot of factors like 1152 does.

Here’s what I’ve learned about the number 1152:

  • 1152 is a composite number.
  • Prime factorization: 1152 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3, which can be written 1152 = 2⁷ × 3²
  • The exponents in the prime factorization are 7 and 2. Adding one to each and multiplying we get (7 + 1)(2 + 1) = 8 × 3 = 24. Therefore 1152 has exactly 24 factors.
  • Factors of 1152: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 128, 144, 192, 288, 384, 576, 1152
  • Factor pairs: 1152 = 1 × 1152, 2 × 576, 3 × 384, 4 × 288, 6 × 192, 8 × 144, 9 × 128, 12 × 96, 16 × 72, 18 × 64, 24 × 48, or 32 × 36
  • Taking the factor pair with the largest square number factor, we get √1152 = (√576)(√2) = 24√2 ≈ 33.94112

The last two digits of 1152 are 52, a number divisible by 4, so 1152 can be evenly divided by 4.
1 + 1 + 5 + 2 = 9, so 1152 is divisible by 9.

It is so easy to tell if a number can be evenly divided by 4 or 9 AND it is so easy to divide by 4 or by 9. When I make a factor cake, I like to see if the current layer of the cake is divisible by 4 or by 9 before I check to see if it is divisible by a prime number.

From that cake, I can quickly tell that 1152 = 2⁷ × 3² by simply counting by 2’s to find the powers of 2 and 3. All the numbers being the same color doesn’t even slow me down.

I can also easily find the √1152 by taking the square root of everything on the outside of the cake:
√1152  = √(4·4) · (√4)(√9)(√2) = (4·2·3)√2 = 24√2

Since MOST square roots that can be simplified are divisible by 4, or by 9, or by both, this is a good strategy to find their square roots.

Here are some more facts about this number:

1152 is the sum of the fourteen prime numbers from 53 to 109,
and it is the sum of the twelve prime numbers from 71 to 127.

34² – 2² = 1152 so we are only 2² = 4 numbers away from the next perfect square.

1152 looks interesting when it is written in these bases:
It’s 800 in BASE 12 because 8(12²) = 1152,
242 in BASE 23 because 2(23²) + 4(23) + 2(1) = 1152
200 in BASE 24 because 2(24²) = 1152,
WW in BASE 35 (W is 32 base 10) because 32(35) + 32(1) = 32(36) = 1152, and
it’s W0 in BASE 36 because 32(36) = 1152

1150 Perfectly Symmetrical Factor Trees

Factor trees can look lovely if they have symmetrical branches. 1150 can make that kind of a tree:

The same number can also make a more disorderly-looking tree:

All of those trees are correct factor trees. And several more can still be made for the number 1150.

What else can I tell you about that number?

  • 1150 is a composite number.
  • Prime factorization: 1150 = 2 × 5 × 5 × 23, which can be written 1150 = 2 × 5² × 23
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1150 has exactly 12 factors.
  • Factors of 1150: 1, 2, 5, 10, 23, 25, 46, 50, 115, 230, 575, 1150
  • Factor pairs: 1150 = 1 × 1150, 2 × 575, 5 × 230, 10 × 115, 23 × 50, or 25 × 46,
  • Taking the factor pair with the largest square number factor, we get √1150 = (√25)(√46) = 5√46 ≈ 33.91165

1150 is the hypotenuse of two Pythagorean triples:
690-920-1150 which is (3-4-5) times 230
322-1104-1150 which is (7-24-25) times 46

1150 looks interesting when it is written in a couple of different bases:
It’s 3232 in BASE 7 because 3(7³) + 2(7²) + 3(7) + 2(1) = 1150
and 6A6 in BASE 13 (A is 10 base 10) because 6(13²) + 10(13) + 6(1) = 1150

1140 is the 18th Tetrahedral Number

1140 is the 18th tetrahedral number because it satisfies this formula:
(18)(18+1)(18+2)/6 = 1140

It is the 18th tetrahedral number because it is the sum of the first 18 triangular numbers:

Since 18 is an even number, 1140 is the sum of the first 9 EVEN squares.

If the 1140 tiny squares in that graphic were cubes, they could be stacked into a tower with either a triangular base or a square base. Then we would see the beauty of this tetrahedral number.

We can see the number 1140 as well as ALL the previous tetrahedral numbers on this portion of Pascal’s Triangle. (They are the green squares.):

1140 has its place as the 3rd number (as well as the 17th number) on the 20th row of Pascal’s triangle because of this next formula:

Here are some other facts about the number 1140:

  • 1140 is a composite number.
  • Prime factorization: 1140 = 2 × 2 × 3 × 5 × 19, which can be written 1140 = 2² × 3 × 5 × 19
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1140 has exactly 24 factors.
  • Factors of 1140: 1, 2, 3, 4, 5, 6, 10, 12, 15, 19, 20, 30, 38, 57, 60, 76, 95, 114, 190, 228, 285, 380, 570, 1140
  • Factor pairs: 1140 = 1 × 1140, 2 × 570, 3 × 380, 4 × 285, 5 × 228, 6 × 190, 10 × 114, 12 × 95, 15 × 76, 19 × 60, 20 × 57, or 30 × 38,
  • Taking the factor pair with the largest square number factor, we get √1140 = (√4)(√285) = 2√285 ≈ 33.76389

Here are some factor trees that use 11 of 1140’s factor pairs:

1140 is the sum of consecutive prime numbers two different ways:
179 + 181 + 191 + 193 + 197 + 199 = 1140,
569 + 571 = 1140

1140 is the hypotenuse of a Pythagorean triple:
684-912-1140 which is (3-4-5) times 228

1140 looks interesting when it is written in a couple other bases:
It’s palindrome 474 in BASE 16 because 4(16²) + 7(16) + 4(1) = 1140,
and it’s 330 in BASE 19 because 3(19²) + 3(19) = 3(19² + 19) = 3(19)(20) = 1140

1134 Some Likely Factor Trees

I could easily make nine factor trees for 1134 using different factor pairs of 1134. However, most people aren’t likely to begin a factor tree by first dividing 1134 by 14, 18, 21, or 27. Most people will want to start with a 1-digit divisor of 1134.

Most people will also use only one color of ink or pencil lead to make a factor tree.

Here are three of 1134’s factor trees that are much more likely to be made by the average person.

How easy is it to find all the prime factors in those one-color trees? You will probably think it is easier for some trees than others.

I think the tree in the middle is the easiest to read. All the prime factors are in the correct order and easy to distinguish from the composite factors in the tree. Although it is similar to using the cake method, I still like the cake method better.

Here are some facts about the number 1134:

  • 1134 is a composite number.
  • Prime factorization: 1134 = 2 × 3 × 3 × 3 × 3 × 7, which can be written 1134 = 2 × 3⁴ × 7
  • The exponents in the prime factorization are 1, 4 and 1. Adding one to each and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 × 5 × 2 = 20. Therefore 1134 has exactly 20 factors.
  • Factors of 1134: 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 81, 126, 162, 189, 378, 567, 1134
  • Factor pairs: 1134 = 1 × 1134, 2 × 567, 3 × 378, 6 × 189, 7 × 162, 9 × 126, 14 × 81, 18 × 63, 21 × 54, or 27 × 42
  • Taking the factor pair with the largest square number factor, we get √1134 = (√81)(√14) = 9√14 ≈ 33.67492

1134 is the sum of four consecutive prime numbers:
277 + 281 + 283 + 293 = 1134

1120 There’s Lots of Shade under These Factor Trees!

It’s hot this summer, but you can rest under the shade of some of 1120’s factor trees. Whether you want a tall tree or a wide one, these are only a few of the MANY possible ones you can choose. Every one of them gives you the same prime factors which are shown in pink here.

Here are some things I’ve learned about the number 1120:

  • 1120 is a composite number.
  • Prime factorization: 1120 = 2 × 2 × 2 × 2 × 2 × 5 × 7, which can be written 1120 = 2⁵ × 5 × 7
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 1120 has exactly 28 factors.
  • Factors of 1120: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 32, 35, 40, 56, 70, 80, 112, 140, 160, 224, 280, 560, 1120
  • Factor pairs: 1120 = 1 × 1120, 2 × 560, 4 × 280, 5 × 224, 7 × 160, 8 × 140, 10 × 112, 14 × 80, 16 × 70, 20 × 56, 28 × 40, or 32 × 35
  • Taking the factor pair with the largest square number factor, we get √1120 = (√16)(√70) = 4√70 ≈ 33.4664

1120 is the sum of the twenty prime numbers from 17 to 101.
It is also the sum of these two consecutive primes:
557 + 563 = 1120

1120 is the hypotenuse of exactly one Pythagorean triple:
672-896-1120 which is (3-4-5) times 224

I like the way 1120 looks in these other bases:
It’s 1112111 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 2(3³) + 3² + 3  + 1 = 1120,
929 in BASE 11 because 9(11²) + 2(11) + 9(1) = 1120,
WW in BASE 34 (W is 32 base 10) because 32(34) + 32(1) = 32(35) = 1120,
and W0 in BASE 35 because 32(35) = 1120