Mathematics

What Kind of Shape is 703 in?

/ ivasallay / 3 Comments
  • 703 is a composite number.
  • Prime factorization: 703 = 19 x 37
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 703 has exactly 4 factors.
  • Factors of 703: 1, 19, 37, 703
  • Factor pairs: 703 = 1 x 703 or 19 x 37
  • 703 has no square factors that allow its square root to be simplified. √703 ≈ 26.514147.

703 is a fascinating number:

Kaprekar number 703

As shown in that graphic, since the square of 703 can be broken apart and added back together to become itself again, 703 is the 7th Kaprekar number. Thank you OEIS.org for that fun fact.

What kind of shape is 703 in?

703 is the 37th triangular number:

703 is 37th triangular number

703 is a triangular number because 37(38)/2 = 703. If we move part of the triangle, those blocks can be neatly rearranged into this rectangle:

703 rectangle

About half of all triangular numbers are also hexagonal numbers. 703 is the 19th hexagonal number because (19⋅2)(19⋅2 – 1)/2 = 703.

In the past I found counting the dots in graphics of larger hexagonal numbers to be a nightmare, but yesterday I saw a small graphic for hexagonal number 28 that used different colors for each expansion. I challenged myself to make a graphic showing that 703 is a hexagonal number. I made it in excel using o’s in different colors. When I finished, I told excel to replace the o’s with •’s. Excel informed me that it made 703 replacements.  Then I adjusted the size of the cells to make the hexagon smaller. I also attempted to make the outer-most hexagon look as symmetrical as possible.

703 is the 19th Hexagonal Number

Hexagonal numbers are not symmetrical the way hexagonal snowflakes are. I thought it would be cool to make 703 into a snowflake, but I didn’t succeed because 703 is just too big and it lacks symmetry. Nevertheless, I must share these directions to make snowflakes that are indeed 6-sided. (Using paper dinner napkins instead of regular paper makes folding and cutting them much simpler.)

703 is a palindrome in three different bases:

  • 383 base 14; note that 3(196) + 8(14) + 3(1) = 703
  • 111 base 26; note that 1(676) + 1(26) + 1(1) =703
  • JJ base 36; note that J is equivalent to 19 in base 10 and 19(36) + 19(1) = 703

Since 37 is one of its factors, 703 is the hypotenuse of the Pythagorean triple we see in this triangle:

703 Hypotenuse

If you divide each number in the triple by 19, you’ll get the primitive Pythagorean triple 12-35-37.

703 is also a leg in some Pythagorean triple triangles:

  • 504-703-865 (a primitive generated from 2(28)(9), 28² – 9², 28² + 9²) Note that 28 + 9 = 37 and 28 – 9 = 19. Both 37 and 19 are prime factors of 703.
  • 703-6660-6697
  • 703-12996-13015
  • 703-247104-247105 (another primitive)

703 Legs

 

 

700 Pick Your Pony! Who will win this Amount of Factors Horse Race?

/ ivasallay / 3 Comments
  • 700 is a composite number.
  • Prime factorization: 700 = 2 x 2 x 5 x 5 x 7, which can be written 700 = (2^2) x (5^2) x 7
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 700 has exactly 18 factors.
  • Factors of 700: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700
  • Factor pairs: 700 = 1 x 700, 2 x 350, 4 x 175, 5 x 140, 7 x 100, 10 x 70, 14 x 50, 20 x 35 or 25 x 28
  • Taking the factor pair with the largest square number factor, we get √700 = (√100)(√7) = 10√7 ≈ 26.457513.

Because this is my 700th post, I think I’ll have another horse race. Some numbers from 601 to 700 have exactly 2 factors, 4 factors, and so forth up to 24 factors. (Only perfect squares can have an odd number of factors.)

Which number from 1 to 24 will win this amount of factors horse race? Which number will come in second place, or third place? Cheering for more than one pony will make the race even more interesting.

Here we see that the numbers 2, 6, & 8 are the first ones out of the gate. Click on the graphic to see the rest of this very thrilling horse race:

605

Every hundred posts I also like to focus on the percentage of numbers whose square roots can be simplified.

700 is divisible by 100 so its square root can easily be simplified: √700 = 10√7.

273 of the first 700 numbers have reducible square roots. That’s exactly 39%.

The rest of the numbers, 427, which is 41% of the first 700 numbers, do not have reducible square roots.

Here’s a table breaking down the amount of factors in each group of one hundred integers and the number of reducible square roots.

1-700 Amount of Factors

Here are some facts about the number 700.

700 is a palindrome in several bases:

  • 4A4 BASE 12; note A is equivalent to 1o in base 10, and 4(144) + 10(12) + 4(1) = 700
  • PP BASE 27; note P is equivalent to 25 in base 10, and 25(27) + 25(1) = 700
  • KK Base 34; note K is equivalent to 20 in base 10, and 20(34) + 20(1) = 700

700 is the sum of four consecutive prime numbers: 167 + 173 + 179 + 181.

Here is a beautiful painting of a horse race that I saw on twitter:

698 The week a single post went viral in two countries

/ ivasallay / 4 Comments

Since this site is called Find the Factors and each post is numbered, I factor those numbers and write some interesting facts about them.

For example, this is my 698th post, so I’ll give its factoring information here:

  • 698 is a composite number.
  • Prime factorization: 698 = 2 x 349
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 698 has exactly 4 factors.
  • Factors of 698: 1, 2, 349, 698
  • Factor pairs: 698 = 1 x 698 or 2 x 349
  • 698 has no square factors that allow its square root to be simplified. √698 ≈ 26.4196896. (Close to the beginning of the square root we see 698 backwards as well as the magic number 689.)

Now I’ll share some other interesting information about the number 698:

698 is the hypotenuse of Pythagorean triple 360-598-698.

OEIS.org informs us that 3^2 + 4^3 + 5^4 = 698.

173 + 174 + 175 + 176 = 698 so it is the sum of four consecutive numbers.

698 is a palindrome in two different bases:

  • 22322 BASE 4; note 2(256) + 2(64) + 3(16) + 2(4) + 2(1) = 698
  • 585 BASE 11; note 5(121) + 8(11) + 5(1) = 698

And 698 is a permutation of 689, the number of my most viewed post of all time. I wrote that post a week ago on Tuesday, and all it contained was some interesting information about the number 689. There was nothing earth shattering in it.

However, Wednesday evening, Denise Gaskins shared that 689th post on her facebook page, Let’s Play Math. (She has so many good things on her facebook page.)

689 Let's Play Math

The 17 shares you see on this facebook post is NOT the whole picture; it is only the beginning.

Other than my homepage/archives, my most viewed post has now become “What Makes 689 Amazing” It has also been shared on facebook over 2000 times!

Top Posts all time

Here is the breakdown of the number of views it received each day:

Number of Views Per Day

Usually most of my views come from the United States and the United Kingdom, but there was a definite although temporary shift last week. Here are the four top counties viewing this blog from last Thursday to Sunday:

Blog Views

I was quite puzzled by why this blog post went viral in Hong Kong and Taiwan. Perhaps Alan Parr’s (wbhs1962) comment about a funny experience with the number 689 broke the ice. Eventually a few readers from Hong Kong and Taiwan enlightened me about the viral interest in this particular number. Those comments were all quite fascinating and an educational experience for me. I invite you to read them as well.

What Makes 689 Amazing?

/ ivasallay / 24 Comments

689 is an amazing number for several reasons. I decided to make graphics to illustrate many of those ways. (689’s factoring information is at the end of this post.)

689 is the sum of consecutive prime numbers 227, 229, and 233.

Also 689 is the sum of the primes from 83 to 109. Do you know what those 7 prime numbers are?

OEIS.org informs us that 689 is the smallest number that can be expressed as the sum of three different square numbers NINE ways. I decided to figure out what those nine ways are and make this first graphic to share with you:

689 Sum of 3 Different Squares

Note: 614 can also be expressed as the sum of 3 squares 9 different ways, but one of those ways is 17² + 17² + 6² = 614, and that duplicates 17² in the same sum.

689 is the same number when it is turned upside down. Numbers with that characteristic are called Strobogrammatic numbers.

689 Rotation

689 BASE 10 isn’t a palindrome, but 373 BASE 14 is; note that 3(196) + 7(14) + 3(1) = 689

Both of 689’s prime factors have a remainder of 1 when divided by 4, so they are hypotenuses of Pythagorean triples. That fact also means 689 can be expressed as the sum of two square numbers TWO different ways, and it makes 689 the hypotenuse of FOUR Pythagorean triples.  Can you tell by looking at the graphic which two are primitive and which two aren’t?

689 Pythagorean triples

689 is the sum of consecutive numbers three different ways:

  • 344 + 345 = 689; that’s 2 consecutive numbers.
  • 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 = 689; that’s 13 consecutive numbers.
  • 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 689; that’s 26 consecutive numbers.

Now you have a few reasons why 689 is an amazing number. 13 and 53 were part of some of those reasons so it shouldn’t surprise anyone to see 13 and 53 pop up in its factoring information, too:

——————————————————————————

  • 689 is a composite number.
  • Prime factorization: 689 = 13 x 53
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 689 has exactly 4 factors.
  • Factors of 689: 1, 13, 53, 689
  • Factor pairs: 689 = 1 x 689 or 13 x 53
  • 689 has no square factors that allow its square root to be simplified. √689 ≈ 26.248809.

685 Is the Sum of Two Squares Two Different Ways

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Because both of its prime factors are hypotenuses of primitive Pythagorean triples, 685 is the hypotenuse of FOUR Pythagorean triples. Two are primitive; two are not:

  • 37-684-685 which was calculated from 19² – 18², 2(19)(18), 19² + 18²
  • 156-667-685 which was calculated from 2(26)(3), 26² – 3², 26² + 3²
  • 411-548-685 (What factor of 685 is the greatest common factor of those 3 numbers?)
  • 440-525-685 (and what is their greatest common factor?)

As you may have notice from those calculations, 685 is the sum of two squares two different ways:

  • 19² + 18² = 685
  • 26² + 3² = 685

685 is the 19th centered square number because 18 and 19 are consecutive numbers and 19² + 18² = 685. There are 685 small squares of various colors in this graphic.

685 is the 19th Centered Squared Number

—————————————————————————————————

  • 685 is a composite number.
  • Prime factorization: 685 = 5 x 137
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 685 has exactly 4 factors.
  • Factors of 685: 1, 5, 137, 685
  • Factor pairs: 685 = 1 x 685 or 5 x 137
  • 685 has no square factors that allow its square root to be simplified. √685 ≈ 26.172505.

—————————————————————————————————

680 What Would Happen If Ten-Frames Looked Like This?

/ ivasallay / 2 Comments

680 is a number made using only even digits. (There’s much more about 680 at the end of the post.)

Numbers ending in 0, 2, 4, 6, or 8 are even. Numbers ending in 1, 3, 5, 7 or 9 are odd. Those two simple concepts are not always easy for young children to understand.

Sometimes we teach rhymes to children to help them know the difference:

  • 0, 2, 4, 6, 8; being EVEN is just great.
  • 1, 3, 5, 7, 9; being ODD is just fine.

Still students in early grades struggle with the concepts of odd and even.

Another seemingly simple concept is what pairs of numbers add up to ten. That concept also isn’t as easy for children to understand as adults might think.

Donna Boucher is an elementary school math interventionist with many years experience. Besides many other topics, she is an expert on teaching adding and subtracting to first and second graders. Here are a couple of her tweets with links to her site:

//platform.twitter.com/widgets.js

and

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Free ten-frame flash cards are available on her site to help students learn addition and subtraction facts. What a powerful way for students to learn! She also has Halloween/Thanksgiving ten-frames for sale at Teachers Pay Teachers.

As I read her post about how to use the ten-frame flash cards I wondered what would happen if we followed her instructions EXACTLY, but the ten-frames looked like this:

7 mitten ten-frame
Children would still learn how to add and subtract, but would they also instinctively learn the difference between odd and even numbers?

Would they figure out for themselves that adding two even numbers or adding two odd numbers ALWAYS makes an even number? Or that adding an odd number and an even number together ALWAYS makes an odd number? Or would changing the ten-frames not make any difference at all? Will the mitten ten-frames only make a difference if the parent/teacher/tutor talks about the odd and even numbers?

I don’t know the answer to those questions, but I think the idea is worth trying. I’ve made Mitten Ten-Frames for all the numbers from 0 to 10. The “empty” frames have outlines of mittens to help children know if a left or a right mitten belongs there. The mitten ten-frames don’t have a second border to guide in cutting them out, so the flashcards might not look as good as Donna Boucher’s, but they should still work as flashcards. Follow Donna Boucher’s instructions exactly. If you use the mitten ten-frames, please add a comment to let me know whether or not they make any difference helping students learn the properties of odd and even numbers.

———————————————————–

Here’s more about the number 680:

1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² = 680.

Because 5, 17, and 85 are some of its factors, 680 is the hypotenuse of four Pythagorean triples. Can you find the greatest common factor of each triple?

  • 104-672-680
  • 288-616-680
  • 320-600-680
  • 408-544-680

680 the 15th tetrahedral number. OEIS.org tells us that it is also the smallest tetrahedral number that can be made by adding two other tetrahedral numbers together, specifically the sum of the 10th and the 14th tetrahedral numbers equals this 15th tetrahedral number as shown below:

  • (10)(11)(12)/6 = 220
  • (14)(15)(16)/6 = 560
  • 220 + 560 = 680
  • (15)(16)(17)/6 = 680

Finally, here is the factoring information for 680:

  • 680 is a composite number.
  • Prime factorization: 680 = 2 x 2 x 2 x 5 x 17, which can be written 680 = (2^3) x 5 x 17
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 680 has exactly 16 factors.
  • Factors of 680: 1, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680
  • Factor pairs: 680 = 1 x 680, 2 x 340, 4 x 170, 5 x 136, 8 x 85, 10 x 68, 17 x 40, or 20 x 34
  • Taking the factor pair with the largest square number factor, we get √680 = (√4)(√170) = 2√170 ≈ 26.0768096.

 

672 Is it too soon to pick out a tree?

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672 can make MANY factor trees.  Here I’ve pictured only a few of the possibilities, one for each of its factor pairs (excluding 1 x 672).

Is it too soon to pick out a tree?

672 Factor Trees

Every one of those trees has the prime factors of 672: 2, 2, 2, 2, 2, 3, and 7, but finding them on each tree might be a challenge because I didn’t distinguish the prime factors from the other factors. Some of those prime factors might seem like they are lost in a pile of leaves. Can you find them on each tree?

672 is the 9th number to have 24 factors. Here is a number line highlighting all nine of those numbers and the distances between them.

24 Factors Number Line

Notice the difference between 672 and the previous number with 24 factors is 12, a record low.

You might get the impression looking at the number line that numbers having 24 factors might be much more common from now on. That may be true, nevertheless, the next number after 672 to have exactly 24 factors is 756 which is 84 numbers away and well past 720 the smallest number to have 30 factors.

Indeed infinitely many numbers have 24 factors, but probably 672 is the last one that will get much attention.

The numbers in the factor pair 24 and 28 are each exactly two numbers away from 26, their average. That means we are just 2² numbers away from 26².

In other words, 672 equals 26² – 2² which can be factored into (26 + 2)(26 – 2) so (26 + 2)(26 – 2) = 28 x 24 = 672.

  • 672 is a composite number.
  • Prime factorization: 672 = 2 x 2 x 2 x 2 x 2 x 3 x 7, which can be written 672 = (2^5) x 3 x 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) = 6 x 2 x 2 = 24. Therefore 672 has exactly 24 factors.
  • Factors of 672: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672
  • Factor pairs: 672 = 1 x 672, 2 x 336, 3 x 224, 4 x 168, 6 x 112, 7 x 96, 8 x 84, 12 x 56, 14 x 48, 16 x 42, 21 x 32, or 24 x 28
  • Taking the factor pair with the largest square number factor, we get √672 = (√16)(√42) = 4√42 ≈ 25.92296.

671 is the Magic Sum of an 11 x 11 Magic Square

/ ivasallay / 3 Comments

6 – 7 + 1 = 0 so 671 is divisible by 11.

671 is the sum of the fifteen prime numbers from 17 to 73.

Because 61 is one of its factors, 671 is the hypotenuse of the Pythagorean triple 121-660-671. The greatest common factor of those three numbers practically jumps out at me. Does it do the same thing to you?

Best of all 671 is the magic sum of an 11 x 11 magic square. (That link from wikipedia helped me construct this square. I’ll give directions so you can do it, too!)

671 Magic Sum for 11 x 11 Magic Square

Notice how every row, column, and diagonal on the square sums to 671. The reason it is the magic sum is because the sum of all the numbers from 1 to 121 can be computed and then divided by 11 (the number of rows). Here is the equation:

  • 671 = 121 x 120/2/11

Because 11 is an odd number there are simple directions to complete the entire square:

The number 1 is located in the exact center of the top row.

Find the number 2 on the square. (It’s located on the bottom row just right of the exact center square.) Notice that the numbers 3, 4, 5, and 6 are on the same diagonal. If you imagine the diagonal wrapping around the square, you can continue to follow it for numbers 7, 8, 9, 10, and 11. We can’t put the number 12 along the same diagonal because the number 1 is already in that spot, so we put the 12 UNDER the 11 and begin working on a new diagonal.

Anytime a number already occupies a space on a diagonal, put the next number under the preceding number and continue making a new diagonal. When a diagonal reaches the edge of the square, imagine that edge is connected to the opposite edge and continue the diagonal from the opposite edge.

I found it to be the trickiest placing the numbers 67 and 68, but other than that it was rather easy to know where to put the numbers.

Notice that the difference between any smaller number and the larger number just below it is either 12 or 1.

If you have excel on your computer, click on 12 Factors 2015-11-02, select the magic square tab, and then you can make this 11 x 11 magic square yourself. As you type in numbers, the columns, rows, and diagonals will automatically keep a running sum.

Once you get the square to give the magic sum in each direction, you can try doing the same thing with the 13 x 13 magic square that I’ve included on the same page. Its magic sum is 1105 which can be also be computed:

  • 1105 = 169 x 170/22/13.

There is actually many more possible and probably more complicated 11 x 11 and 13 x 13 magic squares. I hope you enjoy making some with this easy method.

—————————————————————————————————

  • 671 is a composite number.
  • Prime factorization: 671 = 11 x 61
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 671 has exactly 4 factors.
  • Factors of 671: 1, 11, 61, 671
  • Factor pairs: 671 = 1 x 671 or 11 x 61
  • 671 has no square factors that allow its square root to be simplified. √671 ≈ 25.903668.

—————————————————————————————————

666 The Devil’s Triangular Number

/ ivasallay / 8 Comments

666 is known as the number of the beast. It is also a triangular number because it is the sum of all the counting numbers from 1 to 36.

I felt a math joke was trying to emerge from those two facts so I made a graphic:

666 Is The Devil's Triangular Number

The graphic features 666 blue squares and a few sinking boats traveling through the triangular region between Miami, San Juan, and Bermuda. This region is one of the most traveled throughout the world and is also known as the Devil’s triangle because supposedly more ships have disappeared there than in any other similarly sized body of water.

The sum of the numbers on a roulette wheel is also 666. Adding 0 and 00 to the mix does not change that fact. Even though I was born in Las Vegas and lived in one of its suburbs for over 26 years of my life, I have never spun a roulette wheel or gambled in any other way. This number fact is just one more reason not to do it.

Because 37 is one of its factors, 666 is the hypotenuse of the Pythagorean triple 216-630-666. What is the greatest common factor of those three numbers?

Archimedes-lab lists MANY other interesting facts about the number 666. Here are three of my favorites:

  • In Roman numerals 600 is DC, 60 is LX, and 6 is VI. Thus DCLXVI is 666 and uses all the Roman numeral characters less than M (1000) exactly one time.
  • 15² + 21² = 666; 15 and 21 are both triangular numbers.
  • 2² + 3² + 5² + 7² + 11² + 13² + 17² =666; the numbers being squared are the first seven prime numbers.
  • 666 is the largest repdigit triangular number. It is also a palindrome.

I saw this next number fact on twitter.

//platform.twitter.com/widgets.js

Finally, here is the factoring information for the number 666:

  • 666 is a composite number.
  • Prime factorization: 666 = 2 x 3 x 3 x 37, which can be written 666 = 2 x (3^2) x 37
  • 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 x 3 x 2 = 12. Therefore 666 has exactly 12 factors.
  • Factors of 666: 1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666
  • Factor pairs: 666 = 1 x 666, 2 x 333, 3 x 222, 6 x 111, 9 x 74, or 18 x 37
  • Taking the factor pair with the largest square number factor, we get √666 = (√9)(√74) = 3√74 ≈ 25.8069758

 

658 How Many Triangles Point Up? How Many Triangles Point Down? How Many Triangles in All?

/ ivasallay / 3 Comments

658 Thirteen Rows of Triangles

When I first started counting the triangles in the figure, I started by counting the small triangles:

  • In the top row there is 1 small triangle.
  • In the top 2 rows there are 4 small triangles.
  • In the top 3 rows there are 9 small triangles.
  • In the top 4  rows there are 16 small triangles,
  • and so forth so that in all 13 rows there are 169 small triangles.

As fascinating as that resulting squaring pattern is, it is NOT part of the most efficient way to count ALL the triangles of varying sizes.

The most efficient way to count ALL the triangles can be found on The University of Georgia’s website: Count the triangles that are pointing up separately from the triangles pointing down. Counting charts for triangles with 4, 5, 6, 7, and 8 rows of triangles are displayed on that website. I made a similar chart for these 13 rows of triangles:

658 Chart, Triangles pointed up, Triangles pointed down

Notice all the triangular numbers on the chart!

Because 13 divided by 2 is 6.5, we see that 6 is the largest base size that has any triangles pointing down. Because 6.5 is not a whole number, there are 3 triangles that point down with a base size of 6.

Triangles made from an odd number of rows use these triangular numbers to count the triangles pointing down: 3, 10, 21, 36, 55, 78, etc.

Triangles made from an even number of rows have no remainder when divided by 2 so the triangle with the largest base size is the number of rows divided by two. There will only be one triangle with that base size and the triangular numbers used for that base size and smaller are 1, 6, 15, 28, 45, 66, 91, etc.

The Mathematics Stack Exchange had a discussion on how to count all the triangles, and a formula was posted:

The total number of triangles = ⌊n(n+2)(2n+1)/8⌋ Note: the brackets mean round decimals DOWN to the closest integer.

I made a chart showing the results of using the formula for n = 1 to 13:

658 Formula for number of triangles

Thus we see that 658 is the total number of triangles that can be counted in a triangle made from 13 rows of triangles.

  • 658 is a composite number.
  • Prime factorization: 658 = 2 x 7 x 47
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 658 has exactly 8 factors.
  • Factors of 658: 1, 2, 7, 14, 47, 94, 329, 658
  • Factor pairs: 658 = 1 x 658, 2 x 329, 7 x 94, or 14 x 47
  • 658 has no square factors that allow its square root to be simplified. √658 ≈ 25.65151.