Factors

704 and Level 1

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  • 704 is a composite number.
  • Prime factorization: 704 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 11, which can be written 704 = (2^6) ⋅ 11
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 ⋅ 2 = 14. Therefore 704 has exactly 14 factors.
  • Factors of 704: 1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704
  • Factor pairs: 704 = 1 x 704, 2 x 352, 4 x 176, 8 x 88, 11 x 64, 16 x 44, or 22 x 32
  • Taking the factor pair with the largest square number factor, we get √704 = (√64)(√11) = 8√11 ≈ 26.532998.

This Level 1 puzzle is not difficult at all:

704 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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Here are a few more thoughts about the number 704:

7 – 0 + 4 = 11 so 704 is divisible by 11.

Here’s a fun multiplication fact: 8 x 88 = 704

Also 704 is a palindrome in two different bases:

  • 1I1 in BASE 19; note 19² = 361, I is equivalent to 18 base 10, and 1(361) + 18(19) + 1(1) = 704
  • MM in BASE 31; note M is equivalent to 22 base 10, and 22(31) = 22(1) = 704

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704 Factors

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

 

 

702 A Couple of Christmas Factor Trees

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Since the sum of its digits equals nine, 702 is divisible by nine.

  • 702 is a composite number.
  • Prime factorization: 702 = 2 x 3 x 3 x 3 x 13, which can be written 702 = 2 x (3^3) x 13
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 702 has exactly 16 factors.
  • Factors of 702: 1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702
  • Factor pairs: 702 = 1 x 702, 2 x 351, 3 x 234, 6 x 117, 9 x 78, 13 x 54, 18 x 39, or 26 x 27
  • Taking the factor pair with the largest square number factor, we get √702 = (√9)(√78) = 3√78 ≈ 26.49528.

702 is the product of consecutive integers: 26 x 27 = 702. Numbers that can be expressed as such products are known as Pronic numbers.

It seems only natural to make factor trees based on those two multiplication facts:

702 Factor Trees

Today’s Find the Factors puzzle also looks like a couple of small Christmas trees.

702 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here are more facts about the number 702:

It is the sum of consecutive prime numbers 349 and 353.

It is also the sum of the seventeen prime numbers from 7 to 73.

And because 13 is one of its factors, 702 is the hypotenuse of Pythagorean triple 270-648-702. Notice that the short leg is a permutation of 702.

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702 Logic

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:

Level 4 Christmas Puzzle #699

/ ivasallay / Leave a comment
  • 699 is a composite number.
  • Prime factorization: 699 = 3 x 233
  • 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 699 has exactly 4 factors.
  • Factors of 699: 1, 3, 233, 699
  • Factor pairs: 699 = 1 x 699 or 3 x 233
  • 699 has no square factors that allow its square root to be simplified. √699 ≈ 26.438608.

Here is a Christmas puzzle for you to solve. It’s numbered 699 to distinguish it from every other puzzle I make:

699 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here are a few thoughts I’ve had about the number 699:

699 is the smallest number whose digits add up to 24.

Every odd number greater than 1 is the sum of 2 consecutive numbers. 699 is the sum of 349 and 350.

Every number that is divisible by 3 is the sum of 3 consecutive numbers: 232 + 233 + 234 = 699.

Also 699 is the hypotenuse of Pythagorean triple 315-624-699. Which factor of 699 is the greatest common factor of those three numbers?

699 is palindrome 272 in BASE 17; note that 2(289) + 7(17) + 2(1) = 699

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699 Logic

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.

697 and Level 3

/ ivasallay / Leave a comment
  • 697 is a composite number.
  • Prime factorization: 697 = 17 x 41
  • 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 697 has exactly 4 factors.
  • Factors of 697: 1, 17, 41, 697
  • Factor pairs: 697 = 1 x 697 or 17 x 41
  • 697 has no square factors that allow its square root to be simplified. √697 ≈ 26.40075756.

Here is today’s factoring puzzle:

 

697 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here are some more thoughts about the number 697:

Because both of its factors are hypotenuses of Pythagorean triples, 697 is the hypotenuse of FOUR Pythagorean triples:

  • 153-680-697
  • 185-672-697
  • 328-615-697
  • 455-528-697

Two of those triples are primitive, and two are not. Can you find the greatest common factor for each one that is not primitive?

697 is palindrome 151 in BASE 24; note that 24² = 576, and 1(576) + 5(24) + 1(1) = 697.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

697 Factors

696 There are lots of goodies in this Christmas Stocking

/ ivasallay / Leave a comment
  • 696 is a composite number.
  • Prime factorization: 696 = 2 x 2 x 2 x 3 x 29, which can be written 696 = (2^3) x 3 x 29
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 696 has exactly 16 factors.
  • Factors of 696: 1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348, 696
  • Factor pairs: 696 = 1 x 696, 2 x 348, 3 x 232, 4 x 174, 6 x 116, 8 x 87, 12 x 58, or 24 x 29
  • Taking the factor pair with the largest square number factor, we get √696 = (√4)(√174) = 2√174 ≈ 26.38181.

Today’s puzzle is meant to look like a Christmas stocking or boot that can be filled with lots of little treasures.

696 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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What other facts did I find about the number 696?

696 is the sum of all the prime numbers from 71 to 103. Do you know what those eight prime numbers are?

696 is also the sum of consecutive odd numbers 347 and 349 which just happen to also be consecutive prime numbers.

Because 696 is a multiple of 29, it is the hypotenuse of Pythagorean triple 480-504-696. What is the greatest common factor of those three numbers?

696 is a palindrome in two different bases

  • 696 BASE 10; note that 6(100) + 9(10) + 6(1) = 696
  • OO BASE 28; note that O BASE 28 is equivalent to 24 in BASE 10, and  24(28) + 24(1) = 696

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696 Factors

 

694 and Level 6

/ ivasallay / Leave a comment
  • 694 is a composite number.
  • Prime factorization: 694 = 2 x 347
  • 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 694 has exactly 4 factors.
  • Factors of 694: 1, 2, 347, 694
  • Factor pairs: 694 = 1 x 694 or 2 x 347
  • 694 has no square factors that allow its square root to be simplified. √694 ≈ 26.34388.

Level 6 puzzles aren’t easy, but do give it a try! There are hints at the bottom of the page.

694 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here are some ways that 694 is a special number:

694 is the 22nd centered triangular number because it is the sum of these consecutive triangular numbers 210, 231, 253.

Notice that 21(22)/2 is 231, the number in the middle of those consecutive triangular numbers. Also notice that 3(231) + 1 = 694.

694 BASE 10 becomes palindrome 1C1 in BASE 21 because C Base 21 is equivalent to 12 Base 10, and 1(21²) + 12(21) + 1(1) = 694.

 

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694 Logic

Divisibility Tricks Applied to 693

/ ivasallay / Leave a comment
  • 693 is a composite number.
  • Prime factorization: 693 = 3 x 3 x 7 x 11, which can be written 693 = (3^2) x 7 x 11
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 693 has exactly 12 factors.
  • Factors of 693: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693
  • Factor pairs: 693 = 1 x 693, 3 x 231, 7 x 99, 9 x 77, 11 x 63, or 21 x 33
  • Taking the factor pair with the largest square number factor, we get √693 = (√9)(√77) = 3√77 ≈ 26.324893

Some quick divisibility tricks applied to the number 693:

  1. Every counting number is divisible by 1
  2. 693 is not even so it isn’t divisible by 2
  3. Every digit of 693 is divisible by 3, so 693 is divisible by 3
  4. Since it isn’t divisible by 2, it isn’t divisible by 4
  5. 693 doesn’t end in a 5 or 0, so it’s not divisible by 5
  6. 693 is divisible by 3 but not by 2 so it isn’t divisible by 6
  7. 69 – 2(3) = 63, a multiple of 7 so 693 is divisible by 7
  8. Since it isn’t divisible by 2 or 4, it can’t be divisible by 8
  9. 6 + 9 + 3 = 18, a multiple of 9 so 693 is divisible by 9
  10. Since the last digit of 693 isn’t 0, it is not divisible by 10
  11. 6 – 9 + 3 = 0, so 693 is divisible by 11

The divisibility tricks that worked on the number 693 are quite easy to see on the outside of this factor cake.

Here is today’s factoring puzzle:

693 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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693 is a palindrome in several bases:

  • 1010110101 BASE 2; note 1(512) + 0(256) + 1(128) + 0(64) + 1(32) + 1(16) + 0(8) + 1(4) + 0(2) + 1(1) = 693
  • 3113 BASE 6; note 3(216) + 1(36) + 1(6) + 3(1) = 693
  • 414 BASE 13; note 4(169) + 1(13) + 4(1) = 693
  • 313 BASE 15; note 3(225) + 1(15) + 3(1) = 693

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693 Logic

Ricardo tweeted his work for this puzzle, too.

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