Month: December 2015

708 and Level 5

/ ivasallay / Leave a comment
  • 708 is a composite number.
  • Prime factorization: 708 = 2 x 2 x 3 x 59, which can be written 708 = (2^2) x 3 x 59
  • 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 708 has exactly 12 factors.
  • Factors of 708: 1, 2, 3, 4, 6, 12, 59, 118, 177, 236, 354, 708
  • Factor pairs: 708 = 1 x 708, 2 x 354, 3 x 236, 4 x 177, 6 x 118, or 12 x 59
  • Taking the factor pair with the largest square number factor, we get √708 = (√4)(√177) = 2√177 ≈ 26.608269.

Here is a factor tree for 708:

708 Factor Tree

Today’s factoring puzzle reminds me of lumps of coal:

708 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 facts about 708:

708 can be written as the sum of consecutive numbers three different ways:

  • 235 + 236 + 237 = 708; that’s 3 consecutive numbers.
  • 85 + 86 + 87 + 88 + 89 + 90 + 91 + 92 = 708; that’s 8 consecutive numbers.
  • 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 = 708; that’s 24 consecutive numbers.

708 is palindrome 323 in BASE 15; note that 3(225) + 2(15) + 3(1) = 708.

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708 Logic

707 and Level 4

/ ivasallay / Leave a comment
  • 707 is a composite number.
  • Prime factorization: 707 = 7 x 101
  • 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 707 has exactly 4 factors.
  • Factors of 707: 1, 7, 101, 707
  • Factor pairs: 707 = 1 x 707 or 7 x 101
  • 707 has no square factors that allow its square root to be simplified. √707 ≈ 26.58947.

Here is today’s factoring puzzle:

707 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 facts about 707:

707 is a palindrome in base 10 and in base 100.

707 is the sum of seven consecutive numbers: 98 + 99 + 100 + 101 + 102 + 103 + 104 = 707

It is also the sum of two consecutive numbers: 353 + 354 = 707.

707 is the sum of these five consecutive prime numbers: 131 + 137 + 139 + 149 + 151 = 707.

And it is the sum of the nineteen prime numbers from 5 to 73.

Since 101 is one of its prime factors, 707 is the hypotenuse of Pythagorean triple 140-693-707. What is the greatest common factor of those three numbers?

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707 Logic

706 and Level 3

/ ivasallay / Leave a comment
  • 706 is a composite number.
  • Prime factorization: 706 = 2 x 353
  • 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 706 has exactly 4 factors.
  • Factors of 706: 1, 2, 353, 706
  • Factor pairs: 706 = 1 x 706 or 2 x 353
  • 706 has no square factors that allow its square root to be simplified. √706 ≈ 26.5706

Here is today’s puzzle:

706 Green Red 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 facts about 706:

Because 353 is one of its prime factors, 706 is the hypotenuse of Pythagorean triple 450-544-706. What is the greatest common factor of those three numbers?

Not only is 706 the product of two prime palindromes (2 and 353), but 706 is also a palindrome in 2 different bases:

  • 424 BASE 13; note 4(169) + 2(13) + 4(1) = 706
  • 2C2 BASE 16; note 2(256) + 12(16) + 2(1) = 706

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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 10.)  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.

706 Factors

705 and Level 2

/ ivasallay / Leave a comment
  • 705 is a composite number.
  • Prime factorization: 705 = 3 x 5 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 705 has exactly 8 factors.
  • Factors of 705: 1, 3, 5, 15, 47, 141, 235, 705
  • Factor pairs: 705 = 1 x 705, 3 x 235, 5 x 141, or 15 x 47
  • 705 has no square factors that allow its square root to be simplified. √705 ≈ 26.551836.

Here is today’s puzzle:

 

705 Puzzle

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

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What else can I tell you about the number 705?

Because 5 is one of its factors, 705 is the hypotenuse of the Pythagorean triple 423-564-705. What is the greatest common factor of those three numbers?

705 is the sum of consecutive numbers several different ways:

  • 352 + 353 = 705; (2 consecutive numbers)
  • 234 + 235 + 236 = 705; (3 consecutive numbers)
  • 139 + 140 + 141 + 142 + 143 = 705; (5 consecutive numbers)
  • 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 = 705; (15 consecutive numbers)

705 is palindrome 1A1 in base 22; note that A is equivalent to 10 base 10, 22² = 484, and 1(484) + 10(22) + 1(1) = 705.

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705 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

/ ivasallay / Leave a comment

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

701 Some Virgács left by Mikulás

/ ivasallay / Leave a comment
  • 701 is a prime number.
  • Prime factorization: 701 is prime.
  • The exponent of prime number 701 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 701 has exactly 2 factors.
  • Factors of 701: 1, 701
  • Factor pairs: 701 = 1 x 701
  • 701 has no square factors that allow its square root to be simplified. √701 ≈ 26.4764046.

How do we know that 701 is a prime number? If 701 were not a prime number, then it would be divisible by at least one prime number less than or equal to √701 ≈ 26.4. Since 701 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 701 is a prime number.

Tonight Mikulás will visit the homes of children who sleep in Hungary. If they have been good, he will fill their boots with sweet treasures. If they have been naughty, they will receive virgács, small twigs that have been spray painted gold and bound together with red decorative ribbon. Actually most children make both the naughty list and the nice list so their boots are filled with a mixture of sweet and the not so sweet including virgács, a subtle reminder to be good.

I especially like this illustration from Wikipedia that features Mikulás (Saint Nickolas) and Krampusz:

I like that it is 150 years old. It is from 1865, several decades before any of my husband’s grandparents left Hungary to live in the United States. Under the chair is a little boy hiding from Krampusz. I like to imagine he’s related to my husband some way. The little girl in the illustration must have been much better behaved that year because she is not afraid enough to need to hide.

Since everyone has been at least a little bit naughty this year, here is virgács for you to put in your shoes tonight, too.

701 Puzzle

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

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Here is a little more about the number 701:

26² + 5² = 701 so it is the hypotenuse of the primitive Pythagorean triple 260-651-701 which can be calculated using 2(26)(5), 26² – 5², 26² + 5².

701 is the sum of three consecutive prime numbers: 229 + 233+ 239.

701 is a palindrome in several bases:

  • 10301 BASE 5; note that 1(625) + 0(125) + 3(25) + 0(5) + 1(1) = 701.
  • 858 BASE 9; note that 8(81) + 5(9) + 8(1) = 701.
  • 1F1 BASE 20; note that F is equivalent to 15 in base 10, and 1(400) + 15(20) + 1(1) = 701.
  • 131 BASE 25; note that 1(625) + 3(25) + 1(1) = 701

OEIS.org informs us that 1^0 + 2^1 + 3^2 + 4^3 + 5^4 = 701.

Here’s another way we know that 701 is a prime number: Since 701 ÷ 4 has a remainder of 1, and 701 can be written as the sum of two squares that have no common prime factors (26² + 5² = 701), then 701 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √701 ≈ 26.4. Since 701 is not divisible by 5, 13, or 17, we know that 701 is a prime number.

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701 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