factor pairs

760 Not a Misprint: One side is 84, the other 48. What are the other sides?

/ ivasallay / Leave a comment
  • 760 is a composite number.
  • Prime factorization: 760 = 2 x 2 x 2 x 5 x 19, which can be written 760 = (2^3) x 5 x 19
  • 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 760 has exactly 16 factors.
  • Factors of 760: 1, 2, 4, 5, 8, 10, 19, 20, 38, 40, 76, 95, 152, 190, 380, 760
  • Factor pairs: 760 = 1 x 760, 2 x 380, 4 x 190, 5 x 152, 8 x 95, 10 x 76, 19 x 40, or 20 x 38
  • Taking the factor pair with the largest square number factor, we get √760 = (√4)(√190) = 2√190 ≈ 27.5680975.

760-factor-pairs

There is no misprint in this puzzle. One side really is 84 while the other side really is 48. Can you find the other sides?

760 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s more about the number 760:

760 is the sum of consecutive numbers two different ways. (Two of its factor pairs show up in those ways.):

  • 150 + 151 + 152 + 153 + 154 = 760; that’s 5 consecutive numbers.
  • 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 = 760; that’s 19 consecutive numbers.

760 is the hypotenuse of a Pythagorean triple so that 456² + 608² = 760²

760 is the sum of three squares: 20² + 18² + 6² = 760.

760 is palindrome 1A1 in Base 23 because 1(23²) + 10(23) + 1(1) = 760.

Wikipedia informs us that 760 is the 23rd centered triangular number because (3⋅22² + 3⋅22 + 2)/2 = 760.

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Here’s the same puzzle with out the lists of triples:

760 Puzzle only

759 and Level 6

/ ivasallay / Leave a comment
  • 759 is a composite number.
  • Prime factorization: 759 = 3 x 11 x 23
  • 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 759 has exactly 8 factors.
  • Factors of 759: 1, 3, 11, 23, 33, 69, 253, 759
  • Factor pairs: 759 = 1 x 759, 3 x 253, 11 x 69, or 23 x 33
  • 759 has no square factors that allow its square root to be simplified. √759 ≈ 27.5499546.

759-factor-pairs

Here’s today’s puzzle. A logical way to find its solution is at the end of the post:

759 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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What else can I say about the number 759?

759 can be written as the sum of consecutive numbers seven different ways:

  • 379 + 380 = 759; that’s 2 consecutive numbers.
  • 252 + 253 + 254 = 759; that’s 3 consecutive numbers.
  • 124 + 125 + 126 + 127 + 128 + 129 = 759; that’s 6 consecutive numbers.
  • 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 = 759; that’s 11 consecutive numbers.
  • 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45  = 759; that’s 22 consecutive numbers.
  • 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44  = 759; that’s 23 consecutive numbers.
  • 7 + 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 + 39 = 759; that’s 33 consecutive numbers.

759 is also the sum of five consecutive prime numbers: 139 + 149 + 151 + 157 + 163 = 759.

759 is a palindrome in two bases:

  • 3C3 BASE 14 (C is 12 base 10)
  • NN BASE 32 (N is 23 base 10)

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759 Logic

758 and Level 5

/ ivasallay / Leave a comment
  • 758 is a composite number.
  • Prime factorization: 758 = 2 x 379
  • 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 758 has exactly 4 factors.
  • Factors of 758: 1, 2, 379, 758
  • Factor pairs: 758 = 1 x 758 or 2 x 379
  • 758 has no square factors that allow its square root to be simplified. √758 ≈ 27.5317998.

758-factor-pairs

I started to feel like I was getting behind blogging so I decided over the weekend that it is okay NOT to blog everyday. I didn’t have time to post this puzzle until today, and I have a level 6 puzzle and another whole set of puzzles to post as well. I decided I don’t need to catch up. I’ll post them all soon enough. A logical way to find the solution is in a table at the end of the post.

758 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s more about the number 758:

758 is the sum of four consecutive numbers: 188 + 189 + 190 + 191 = 758.

758 is the sum of three squares six different ways:

  • 27² + 5² + 2² = 758
  • 26² + 9² + 1² = 756
  • 23² + 15² + 2² = 758
  • 22² + 15² + 7² = 758
  • 21² + 14² + 11² = 758
  • 19² + 19² + 6² = 758

758 is a palindrome in two different bases:

  • 464 BASE 13; note that 4(13²) + 6(13) + 4(1) = 758.
  • 262 BASE 18; note that 2(18²) + 6(18) + 2(1) = 758.

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758 Logic

 

 

756 and Level 3

/ ivasallay / Leave a comment

Today’s Puzzle:

756 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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.

756 Factors

Factor Trees for 756:

756 has many factors and, therefore, it has many possible factor trees. Here are three of them:

756 factor trees

Factors of 756:

  • 756 is a composite number.
  • Prime factorization: 756 = 2 x 2 x 3 x 3 x 3 x 7, which can be written 756 = 2² x 3³ x 7
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 756 has exactly 24 factors.
  • Factors of 756: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756
  • Factor pairs: 756 = 1 x 756, 2 x 378, 3 x 252, 4 x 189, 6 x 126, 7 x 108, 9 x 84, 12 x 63, 14 x 54, 18 x 42, 21 x 36 or 27 x 28
  • Taking the factor pair with the largest square number factor, we get √756 = (√21)(√36) = 6√21 ≈ 27.495454.

756-factor-pairs

Sum-Difference Puzzles:

84 has six factor pairs. One of those pairs adds up to 25, and another one subtracts to 25. Put the factors in the appropriate boxes in the first puzzle.

756 has twelve factor pairs. One of the factor pairs adds up to ­75, and a different one subtracts to 75. 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 756:

The last two digits of 756 is divisible by 4 so 756 is divisible by 4.

756 is formed from 3 consecutive numbers (5, 6, 7) so it is divisible by 3. The middle number is divisible by 3 so 756 is also divisible by 9.

756 can be written as the sum of consecutive numbers seven ways:

  • 251 + 252 + 253 = 756; that’s 3 consecutive numbers.
  • 105 + 106 + 107 + 108 + 109 + 110 + 111 = 756; that’s 7 consecutive numbers.
  • 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 = 756; that’s 8 consecutive numbers.
  • 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 756; that’s 9 consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 756; that’s 21 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 + 43 = 756; that’s 24 consecutive numbers.
  • 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 + 40 + 41  = 756; that’s 27 consecutive numbers.

756 is also the sum of six consecutive prime numbers: 109 + 113 + 127+ 131 + 137+ 139 = 756.

756 can be written as the sum of three squares four different ways. (Notice that all of the squares are even):

  • 26² + 8² + 4² = 756
  • 24² + 12² + 6² = 756
  • 22² + 16² + 4² = 756
  • 20² + 16² + 10² = 756

756 is a palindrome in two other bases:

  • 11011 BASE 5; note that 1(625) + 1(125) + 0(25) + 1(5) + 1(1) = 756.
  • LL BASE 35 (L is 21 base 10); note that 21(35) + 21(1) = 756.

 

755 and Level 2

/ ivasallay / Leave a comment
  • 755 is a composite number.
  • Prime factorization: 755 = 5 x 151
  • 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 755 has exactly 4 factors.
  • Factors of 755: 1, 5, 151, 755
  • Factor pairs: 755 = 1 x 755 or 5 x 151
  • 755 has no square factors that allow its square root to be simplified. √755 ≈ 27.4772633.

755-factor-pairs

Here’s today’s puzzle:

 

755 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s a little more about 755:

755 is the sum of consecutive numbers three different ways:

  • 377 + 378 = 755; that’s 2 consecutive numbers.
  • 149 + 150 + 151 + 152 + 153 = 755; that’s 5 consecutive numbers.
  • 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 = 755; that’s 10 consecutive numbers.

Because 5 is one of its factors, 755 is the hypotenuse of Pythagorean triple 453-604-755.

755 is the sum of three squares six different ways:

  • 27² + 5² + 1² = 755
  • 25² + 11² + 3² = 755
  • 25² + 9² + 7² = 755
  • 23² + 15² + 1² = 755
  • 21² + 17² + 5² = 755
  • 19² + 15² + 13² = 755

755 is palindrome 131 in BASE 26; note that 1(26²) + 3(26) + 1(1) = 755.

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755 Factors

754 and Level 1

/ ivasallay / Leave a comment
  • 754 is a composite number.
  • Prime factorization: 754 = 2 x 13 x 29
  • 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 754 has exactly 8 factors.
  • Factors of 754: 1, 2, 13, 26, 29, 58, 377, 754
  • Factor pairs: 754 = 1 x 754, 2 x 377, 13 x 58, or 26 x 29
  • 754 has no square factors that allow its square root to be simplified. √754 ≈ 27.459060.

754-factor-pairs

Here’s today’s puzzle:

754 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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I have much more to tell you about the number 754:

754 is the sum of consecutive numbers three different ways:

  • 187 + 188 + 189 + 190 = 754; that’s 4 consecutive numbers.
  • 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 = 754; that’s 13 consecutive numbers.
  • 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 + 39 + 40 = 754; that’s 29 consecutive numbers.

Because all of the odd prime factors of 754 can be expressed as the sum of two squares, 754 can also be expressed as the sum of two squares:

  • 27² + 5² = 754
  • 23² + 15² = 754

Also because two of its prime factors are hypotenuses of primitive Pythagorean triples, I knew that 754 is the hypotenuse of FOUR Pythagorean triples:

  • 270² + 704² = 754²; the triple 270-704-754 was calculated from 2(27)(5), 27² – 5², 27² + 5².
  • 290² + 696² = 754²
  • 304² + 690² = 754²; the triple 304-690-754 was calculated from 23² – 15², 2(23)(15), 23² + 15².
  • 520² + 546² = 754²

754 can also be written as the sum of three squares four different ways:

  • 27² + 4² + 3² = 754
  • 24² + 13² + 3² = 754
  • 23² + 12² + 9² = 754
  • 21² + 13² + 12² = 754

754 is a palindrome in three bases:

  • 626 BASE 11; note that 6(121) + 2(11) + 6(1) = 754.
  • 2F2 BASE 16 (F is 15 base 10); note that 2(256) + 15(16) + 2(1) = 754.
  • QQ BASE 28(Q is 26 base 10); note that 26(28) + 26(1) = 754.

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754 Factors

753 Pythagorean Triple Puzzle

/ ivasallay / Leave a comment
  • 753 is a composite number.
  • Prime factorization: 753 = 3 x 251
  • 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 753 has exactly 4 factors.
  • Factors of 753: 1, 3, 251, 753
  • Factor pairs: 753 = 1 x 753 or 3 x 251
  • 753 has no square factors that allow its square root to be simplified. √753 ≈ 27.440845.

753-factor-pairs

PUZZLE DIRECTIONS: This puzzle is NOT drawn to scale. Angles that are marked as right angles are 90 degrees, but any angle that looks like a 45 degree angle, isn’t 45 degrees. Lines that look parallel are NOT parallel. Shorter looking line segments may actually be longer than longer looking line segments. Most rules of geometry do not apply here: in fact non-adjacent triangles in the drawing might actually overlap.

No geometry is needed to solve this puzzle. All that is needed is the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.

753 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

Sorted Triples

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Here are some fun facts about the number 753:

753 is made from three consecutive odd numbers so it is divisible by 3.

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

  • 376 + 377 = 753; that’s 2 consecutive numbers.
  • 250 + 251 + 252 = 753; that’s 3 consecutive numbers.
  • 123 + 124 + 125 + 126 + 127 + 128 = 753; that’s 6 consecutive numbers.

753 can be written as the sum of three cubes four ways, two of which duplicate squares:

  • 25² + 8² + 8² = 753
  • 22² + 13² + 10² = 753
  • 20² + 17² + 8² = 753
  • 19² + 14² + 14² = 753

From OEIS.org we learn that 753^3 = 426,957,777, the smallest positive perfect cube to contain 4 consecutive 7’s.

753 is palindrome 353 in BASE 15; note that 3(225) + 5(15) + 3(1) = 753.

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752 and Level 6

/ ivasallay / Leave a comment
  • 752 is a composite number.
  • Prime factorization: 752 = 2 x 2 x 2 x 2 x 47, which can be written 752 = (2^4) x 47
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 752 has exactly 10 factors.
  • Factors of 752: 1, 2, 4, 8, 16, 47, 94, 188, 376, 752
  • Factor pairs: 752 = 1 x 752, 2 x 376, 4 x 188, 8 x 94, or 16 x 47
  • Taking the factor pair with the largest square number factor, we get √752 = (√16)(√47) = 4√47 ≈ 27.422618.

752-factor-pairs

Here’s today’s puzzle. A logical way to solve it is given in the table at the end of the post.

752 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

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Here’s a few more thoughts about the number 752:

52 is divisible by 4 so 752 is also divisible by 4. However, 52 is not also divisible by 8, but since 7 is odd, 752 IS divisible by 8.

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 + 39 = 752; that’s 32 consecutive numbers.

752 is the sum of two consecutive primes: 373 + 379 = 752.

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752 Logic

750 and Level 4

/ ivasallay / Leave a comment

Today’s Puzzle:

750 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

Some possible steps to solve the puzzle:

750 Logic

Factors of 750:

  • 750 is a composite number.
  • Prime factorization: 750 = 2 x 3 x 5 x 5 x 5, which can be written 750 = 2 x 3 x (5^3)
  • The exponents in the prime factorization are 1, 1, and 3. Adding one to each and multiplying we get (1 + 1)(1 + 1)(3 + 1) = 2 x 2 x 4 = 16. Therefore 750 has exactly 16 factors.
  • Factors of 750: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750
  • Factor pairs: 750 = 1 x 750, 2 x 375, 3 x 250, 5 x 150, 6 x 125, 10 x 75, 15 x 50, or 25 x 30
  • Taking the factor pair with the largest square number factor, we get √750 = (√25)(√30) = 5√30 ≈ 27.386127875.

750-factor-pairs

Sum-Difference Puzzles:

30 has four factor pairs. One of those pairs adds up to 13, and another one subtracts to 13. Put the factors in the appropriate boxes in the first puzzle.

750 has eight factor pairs. One of the factor pairs adds up to ­65, and a different one subtracts to 65. 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 reasons to be interested in the number 750:

750 can be written as the sum of consecutive numbers seven ways:

  • 249 + 250 + 251 = 750; that’s 3 consecutive numbers.
  • 186 + 187 + 188 + 189 = 750; that’s 4 consecutive numbers.
  • 148 + 149 + 150 + 151 + 152 = 750; that’s 5 consecutive numbers.
  • 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 = 750; that’s 12 consecutive numbers.
  • 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 = 750; that’s 15 consecutive numbers.
  • 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 750; that’s 20 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 + 42 = 750; that’s 25 consecutive numbers.

750 is also the sum of all the prime numbers from 17 to 79. Do you remember what those sixteen prime numbers are?

Because 5, 25, and 125 are its factors, 750 is the hypotenuse of three Pythagorean triple triangles:

  • 210² + 720² = 750²
  • 264² + 702² = 750²
  • 450² + 600² = 750²

750² = 562500 which is another cool looking square number whose digits include 5 and the value of 5^4.

750 is also the sum of three squares six different ways:

  • 26² + 7² + 5² = 750
  • 25² + 11² + 2² = 750
  • 25² + 10² + 5² = 750
  • 23² + 14² + 5² = 750
  • 23² + 11² + 10² = 750
  • 19² + 17² + 10² = 750

Wikipedia tells us that 750 is the 15th nonagonal number because 15(7⋅15 – 5)/2 = 750. It is also 10 times the 5th nonogonal number because 10 ⋅ 5(7⋅5 – 5)/2 = 750.

750 is a palindrome in three different bases:

  • 23232 BASE 4; note that 2(4^4) + 3(4^3) + 2(4^2) + 3(4^1) + 2(4^0) = 750.
  • 2A2 BASE 17 (A= 10 base 10); note that 2(17²) + 10(17) + 2(1) = 750.
  • PP BASE 29 (P = 25 base 10); note that 25(29) + 25(1) = 750.

749 and Level 3

/ ivasallay / Leave a comment

Obviously 749 can be evenly divided by 7.

  • 749 is a composite number.
  • Prime factorization: 749 = 7 x 107
  • 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 749 has exactly 4 factors.
  • Factors of 749: 1, 7, 107, 749
  • Factor pairs: 749 = 1 x 749 or 7 x 107
  • 749 has no square factors that allow its square root to be simplified. √749 ≈ 27.367864.

749-factor-pairs

Here’s today’s puzzle:

749 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

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Here’s a little more about the number 749:

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

  • 374 + 375 = 749; that’s 2 consecutive numbers.
  • 104 + 105 + 106 + 107 + 108 + 109 + 110 = 749; that’s 7 consecutive numbers.
  • 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 = 749; that’s 14 consecutive numbers.

749 is the sum of three consecutive prime numbers: 241 + 251 + 257 = 749.

749 is the sum of three cubes eight different ways:

  1. 27² + 4² + 2² = 749
  2. 26² + 8² + 3² = 749
  3. 24² + 13² + 2² = 749
  4. 22² + 16² + 3² = 749
  5. 22² + 12² + 11² = 749
  6. 20² + 18² + 5² = 749
  7. 19² + 18² + 8² = 749
  8. 18² + 16² + 13² = 749

749 is a palindrome in two different bases:

  • 525 BASE 12; note that 5(144) + 2(12) + 5(1) = 749
  • 1C1 BASE 22 (C = 12 base 10); note that 1(22²) + 12(22) + 1(1) = 749

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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.

749 Factors