## A Multiplication Based Logic Puzzle

### 757 and Level 4

• 757 is a prime number.
• Prime factorization: 757 is prime and cannot be factored.
• The exponent of prime number 757 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 757 has exactly 2 factors.
• Factors of 757: 1, 757
• Factor pairs: 757 = 1 x 757
• 757 has no square factors that allow its square root to be simplified. √757 ≈ 27.51363.

How do we know that 757 is a prime number? If 757 were not a prime number, then it would be divisible by at least one prime number less than or equal to √757 ≈ 27.5. Since 757 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 757 is a prime number.

Here’s today’s puzzle. A logical way to solve it can be found in a table at the end of the post.

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

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Here’s another way we know that 757 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 9² = 757 with 26 and 9 having no common prime factors, 757 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √757 ≈ 27.5. Since 757 is not divisible by 5, 13, or 17, we know that 757 is a prime number.

757 is prime so the only way it can be written as the sum of consecutive numbers is 378 + 379 = 757.

757 is the sum of seven consecutive prime numbers: 97 + 101 + 103 + 107 + 109 + 113 + 127 = 757.

From Stetson.edu we learn this distinguishing fact about the number 757:

Here are some square facts about the number 757:

26² + 9² = 757, and 468² + 595² = 757². That primitive Pythagorean triple, 468-595-757, can be calculated from 2(26)(9), 26² – 9², 26² + 9².

757 is also the sum of three squares two different ways:

• 24² + 10² + 9² = 757
• 18² + 17² + 12² = 757

Finally 757 is a palindrome in four different bases:

• 1001001 BASE 3; note that 1(3^6) + 0(3^5) + 0(3^4) + 1(3^3) + 0(3^2) + 0(3^1) + 1(3^0) = 757.
• 757 BASE 10; note that 7(100) + 5(10) + 7(1) = 757.
• 1F1 BASE 21 (F is 15 base 10); note that 1(21²) + 15(21) + 1(1) = 757.
• 111 BASE 27; note that 1(27²) + 1(27) + 1(1) = 757.

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### 756 and Level 3

• 756 is a composite number.
• Prime factorization: 756 = 2 x 2 x 3 x 3 x 3 x 7, which can be written 756 = (2^2) x (3^3) 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 has many factors and, therefore, it has many possible factor trees. Here are three of them:

Here’s a level 3 Find the Factors puzzle for you to solve, too:

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

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Here’s a few more thoughts on 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.

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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.

### 755 and Level 2

• 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.

Here’s today’s 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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### 754 and Level 1

• 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.

Here’s today’s 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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### 753 Pythagorean Triple Puzzle

• 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.

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.

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

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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 Stetson.edu 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

• 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.

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

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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### 751 and Level 5

• 751 is a prime number.
• Prime factorization: 751 is prime and cannot be factored.
• The exponent of prime number 751 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 751 has exactly 2 factors.
• Factors of 751: 1, 751
• Factor pairs: 751 = 1 x 751
• 751 has no square factors that allow its square root to be simplified. √751 ≈ 27.404379.

How do we know that 751 is a prime number? If 751 were not a prime number, then it would be divisible by at least one prime number less than or equal to √751 ≈ 27.4. Since 751 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 751 is a prime number.

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

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

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Here’s two more thoughts about prime number 751.

Since 751 is a prime number, there is only one way it can be written as the sum of consecutive numbers: 375 + 376 = 751.

Also 751 is palindrome 151 in BASE 25; note that 1(625) + 5(25) +1(1) = 751.

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