factor pairs

722 and Level 5

/ ivasallay / Leave a comment
  • 722 is a composite number.
  • Prime factorization: 722 = 2 x 19 x 19, which can be written 722 = 2 x (19^2)
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 722 has exactly 6 factors.
  • Factors of 722: 1, 2, 19, 38, 361, 722
  • Factor pairs: 722 = 1 x 722, 2 x 361, or 19 x 38
  • Taking the factor pair with the largest square number factor, we get √722 = (√361)(√2) = 19√2 ≈ 26.870058.

Factoring 722 is a lot easier if you’ve memorized that 19² = 361.

Here is today’s factoring puzzle:

722 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here are some other interesting facts about the number 722:

I learned from OEIS.org that (2^4) + (3^4) + (5^4) = 722. The red numbers are the first 3 prime numbers.

722 is also the sum of consecutive numbers two different ways:

  • 179 + 180 + 181 + 182 = 722; that’s 4 consecutive numbers.
  • 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39+ 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 722; that’s 19 consecutive numbers.

722 is a palindrome in two other bases:

  • 2D2 BASE 16 (D = 13 base 10); note that 2(16²) + 13(16) + 2(1) = 722.
  • 242 BASE 18; note that 2(18²) + 4(18) + 2(1) = 722.

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722 Logic

721 Merry Christmas, Everybody!

/ ivasallay / 2 Comments

It isn’t difficult to see that 721 is divisible by 7.

  • 721 is a composite number.
  • Prime factorization: 721 = 7 x 103
  • 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 721 has exactly 4 factors.
  • Factors of 721: 1, 7, 103, 721
  • Factor pairs: 721 = 1 x 721 or 7 x 103
  • 721 has no square factors that allow its square root to be simplified. √721 ≈ 26.85144.

Merry Christmas everybody! Today’s puzzle is not too hard and not too easy so enjoy solving it during your leisure hours today.

721 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here is a little more about the number 721:

721 is the sum of the nine prime numbers from 61 to 101.

OEIS.org informs us that 721 can be expressed as the difference of two cubes two different ways, and is the smallest number that can make that claim. The two ways were fairly easy to find:

  • 9^3 – 2^3 = 721
  • 16^3 – 15^3 = 721

Because it is equal to the difference of the 16th and 15th cubes, 721 is the 16th centered hexagonal number.

And 721 is a palindrome in two bases:

  • 1G1 in base 20 (G = 16 base 10); note that 1(400) + 16(20) + 1(1) = 721.
  • 161 base 24; note that 1(24²) + 6(24) + 1(1) = 721.

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721 Logic

720 Christmas Factor Trees

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Possible Christmas Dinner Choices:

Since I hosted a Christmas Eve dinner for my extended family, I thought of some of the many ways people could fill up their plates.

720 Dinners

If we made a tree diagram of all the possible dinners that could be made with exactly one item from each column, it would contain 720 lines and require quite a few pages.

The fundamental counting principle tells us the easiest way to count all those dinners is to multiply together the number of items in each column. In this case that would be 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 720.

6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 720, and is usually written 6! = 720.

Factor Trees for 720:

Making a tree diagram for 720 would be quite tedious, but making a factor tree for 720 is so easy, that I made several of them:

720 Some Factor Trees

None of those were very attractive, but here are some that are much better looking and actually look like Christmas trees:

720 More Factor Trees

Factors of 720:

720 has 30 factors. That’s more than any previous number!

  • 720 is a composite number.
  • Prime factorization: 720 = 2 x 2 x 2 x 2 x 3 x 3 x 5, which can be written 720 = (2^4) x (3^2) x 5
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each and multiplying we get (4 + 1)(2 + 1)(1 + 1) = 5 x 3 x 2 = 30. Therefore 720 has exactly 30 factors.
  • Factors of 720: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720
  • Factor pairs: 720 = 1 x 720, 2 x 360, 3 x 240, 4 x 180, 5 x 144, 6 x 120, 8 x 90, 9 x 80, 10 x 72, 12 x 60, 15 x 48, 16 x 45, 18 x 40, 20 x 36 or 24 x 30
  • Taking the factor pair with the largest square number factor, we get √720 = (√144)(√5) = 12√5 ≈ 26.8328157.

Sum-Difference Puzzles:

180 has nine factor pairs. One of those pairs adds up to 41, and  another one subtracts to 41. Put the factors in the appropriate boxes in the first puzzle.

720 has fifteen factor pairs. One of the factor pairs adds up to ­82, and a different one subtracts to 82. 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?

More about the Number 720:

Here are some other random facts about the number 720:

Because 5 is one of its factors, 720 is the hypotenuse of the Pythagorean triple 432-576-720. What is the greatest common factor of those 3 numbers? The greatest common factor is in the factor pair with the number 5. All 15 factor pairs for 720 are listed at the end of the post.

720 Hypotenuse

The sum of the interior angles of any hexagon is 720 degrees.

720 degrees hexagons

720 is a palindrome in three different bases:

  • 5A5 in base 11; note that 5(121) + 10(11) + 5(1) = 720.
  • OO in base 29; (O is 24 base 10) note that 24(29) + 24(1) = 720.
  • KK in base 35; (K is 20 base 10) note that 20(35) + 20(1) = 720.

 

718 and a Christmas Star

/ ivasallay / Leave a comment
  • 718 is a composite number.
  • Prime factorization: 718 = 2 x 359
  • 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 718 has exactly 4 factors.
  • Factors of 718: 1, 2, 359, 718
  • Factor pairs: 718 = 1 x 718 or 2 x 359
  • 718 has no square factors that allow its square root to be simplified. √718 ≈ 26.795522.

Today’s puzzle looks like a Christmas Star.

718 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here’s a few facts about the number 718:

178 + 179 + 180 + 181 = 718 making it the sum of 4 consecutive numbers.

718 is the sum of three squares three different ways:

  • 3² + 15² + 22² = 718
  • 9² + 14² + 21² = 718
  • 13² + 15² + 18² = 718

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718 Factors

717 and Level 1

/ ivasallay / Leave a comment
  • 717 is a composite number.
  • Prime factorization: 717 = 3 x 239
  • 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 717 has exactly 4 factors.
  • Factors of 717: 1, 3, 239, 717
  • Factor pairs: 717 = 1 x 717 or 3 x 239
  • 717 has no square factors that allow its square root to be simplified. √717 ≈ 26.77685567799.

If you know a few square numbers, this puzzle is pretty easy:

717 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here are a few more facts I’ve found about the number 717:

717 can be written as the sum of consecutive numbers 2 ways:

  • 358 + 359 = 717; that’s 2 consecutive numbers.
  • 238 + 239 + 240 = 717; that’s 3 consecutive numbers.

717 is a palindrome in three bases:

  • 1011001101 BASE 2
  • 717 BASE 10
  • 393 BASE 14

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717 Factors

716 and Level 6

/ ivasallay / Leave a comment
  • 716 is a composite number.
  • Prime factorization: 716 = 2 x 2 x 179, which can be written 716 = (2^2) x 179
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 716 has exactly 6 factors.
  • Factors of 716: 1, 2, 4, 179, 358, 716
  • Factor pairs: 716 = 1 x 716, 2 x 358, or 4 x 179
  • Taking the factor pair with the largest square number factor, we get √716 = (√4)(√179) = 2√179 ≈ 26.758176.

You may find this Candy Cane puzzle more challenging than sweet:

 

716 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Here are two more facts about the number 716:

86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 = 716; that’s 8 consecutive numbers.

716 is palindrome 282 in BASE 17; note that 2(17²) + 8(17) + 2(1) = 716

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716 Logic

715 and Level 5

/ ivasallay / Leave a comment
  • 715 is a composite number.
  • Prime factorization: 715 = 5 x 11 x 13
  • 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 715 has exactly 8 factors.
  • Factors of 715: 1, 5, 11, 13, 55, 65, 143, 715
  • Factor pairs: 715 = 1 x 715, 5 x 143, 11 x 65, or 13 x 55
  • 715 has no square factors that allow its square root to be simplified. √715 ≈ 26.73948.

Can you solve today’s puzzle?

715 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Here are several more facts about the number 715:

715 is the 22nd pentagonal number because 22(3⋅22 – 1)/2 = 715.

715 can be written as the sum of consecutive numbers several ways:

  • 357 +358 = 715; that’s 2 consecutive numbers.
  • 141 + 142 + 143 + 144 + 145 = 715; that’s 5 consecutive numbers.
  • 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 = 715; that’s 11 consecutive numbers.
  • 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 = 715; that’s 13 consecutive numbers.

Since 5 and 13 are two of its factors, 715 is the hypotenuse of four Pythagorean triples.

  • 176-693-715
  • 275-660-715
  • 363-616-715
  • 429-572-715

Each of those triples has a greatest common factor. Can you find them all?

715 is also palindrome 1D1 in BASE 21 where D = 13 base 10; note that 1(21²) + 13(21) + 1(1) = 715.

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715 Logic

714 Home Runs

/ ivasallay / 4 Comments
  • 714 is a composite number.
  • Prime factorization: 714 = 2 x 3 x 7 x 17
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 714 has exactly 16 factors.
  • Factors of 714: 1, 2, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714
  • Factor pairs: 714 = 1 x 714, 2 x 357, 3 x 238, 6 x 119, 7 x 102, 14 x 51, 17 x 42, or 21 x 34
  • 714 has no square factors that allow its square root to be simplified. √714 ≈ 26.720778.

Babe Ruth hit 714 home runs. His record for the most home runs stood for nearly 39 years when Hank Aaron hit the 715th home run of his career.

The number of prime factors that 714 has is a power of two so 714 can make a very attractive factor tree all by itself. 715 only has three prime factors so its factor tree doesn’t look as nice. I still wanted to make factor trees for both of them so I multiplied them together and made this impressive looking factor tree:

714 and 715 Baseballs and Factor Trees

(The picture of the 714 baseball in the graphic above listed e-bay as the source while the picture of the 715 baseball was from Sports Memorabilia.com. The 715 ball is autographed by Al Downing, the former all-star and comeback player of the year, who pitched the ball that became Hank Aaron’s 715th home run.)

The numbers 714 and 715 form what is now called a Ruth Aaron Pair. Other consecutive numbers whose factors add up to equal numbers are also given that name.

Between the numbers 714 and 715 ALL the prime numbers less than or equal to 17 are used as factors exactly one time.

I became aware of these facts several months ago when Dan Bach sent me this tweet:

//platform.twitter.com/widgets.js

This is my 714th post so I will focus on the number 714 and some different ways it hits a home run:

714 can be written as the sum of consecutive numbers many ways:

  • 237 + 238 +239 = 714; that’s 3 consecutive numbers.
  • 177 + 178 + 179 + 180 = 714; that’s 4 consecutive numbers.
  • 99 + 100 + 101 + 102 + 103 + 104 + 105 = 714; that’s 7 consecutive numbers.
  • 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 = 714; that’s 12 consecutive numbers.
  • 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 = 714; that’s 17 consecutive numbers.
  • 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 = 714; that’s 21 consecutive numbers.

714 is the sum of the twelve prime numbers from 37 to 83.

714 is also the sum of the fourteen prime numbers from 23 to 79. Can you name all those prime numbers?

Because 17 is one of its factors, 714 is the hypotenuse of the Pythagorean triple 336-630-714. What is the greatest common factor of those three numbers?

OEIS.org alerted me to the fact that 714 is the smallest number that uses all possible digits exactly the same number of times in both BASE 2 and BASE 5:

  • 714 is 1011001010 in BASE 2; that’s five 1’s and five 0’s, the only digits used in BASE 2.
  • 714 is 10324 in BASE 5; note that 0, 1, 2, 3, and 4 are the only digits used in BASE 5, and each is used exactly once.

In addition 714 is a palindrome in two bases:

  • 181 BASE 23; note that 1(23²) + 8(23) + 1(1) = 714.
  • LL BASE 33 (L is 21 in base 10); note that 21(33) + 21(1) = 714.

 

After I wrote this post, I tweeted Dan Bach about it, and he let me know how he first learned about Ruth Aaron pairs:

//platform.twitter.com/widgets.js

713 and Level 4

/ ivasallay / Leave a comment
  • 713 is a composite number.
  • Prime factorization: 713 = 23 x 31
  • 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 713 has exactly 4 factors.
  • Factors of 713: 1, 23, 31, 713
  • Factor pairs: 713 = 1 x 713 or 23 x 31
  • 713 has no square factors that allow its square root to be simplified. √713 ≈ 26.7020598.

Go ahead and give this Level 4 puzzle a try:

 

713 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Would you like to know a little more about 713?

713 can be written as the sum of consecutive numbers several ways:

  • 356 + 357 = 713; that’s two consecutive numbers.
  • 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 713; that’s 23 consecutive numbers.
  • 8 + 9 + 10 + 11 + 12 + 13 + 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 = 713; that’s 31 consecutive numbers.

713 is also the sum of three consecutive prime numbers: 233 + 239 + 241 = 713.

Two of its factors are 4 numbers away from their average, 27, so (23)(31) = (27 – 4)(27 + 4) = 27² – 4² = 713

713 is palindrome NN in BASE 30 (N = 23 in base 10); note that 23(30) +23(1) = 713.

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713 Logic

Thank you, Ricardo, for tweeting the puzzle and your work:

//platform.twitter.com/widgets.js

712 and Level 3

/ ivasallay / Leave a comment
  • 712 is a composite number.
  • Prime factorization: 712 = 2 x 2 x 2 x 89, which can be written 712 = (2^3) x 89
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 712 has exactly 8 factors.
  • Factors of 712: 1, 2, 4, 8, 89, 178, 356, 712
  • Factor pairs: 712 = 1 x 712, 2 x 356, 4 x 178, or 8 x 89
  • Taking the factor pair with the largest square number factor, we get √712 = (√4)(√178) = 2√178 ≈ 26.683328.

Here’s a Level 3 puzzle for you to solve:

 

712 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Here are a few more facts about the number 712:

37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 = 712; that’s 16 consecutive numbers.

712 is the sum of the twenty-one prime numbers from 2 to 73.

It is also the sum of consecutive prime numbers 353 and 359.

Because 89 is one of its prime factors, 712 is the hypotenuse of the Pythagorean triple 312-640-712. What is the greatest common factor of those three numbers?

712 is palindrome 4B4 in BASE 12 because 4(144) + 11(12) + 4(1) = 712.

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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.

712 Factors