A Multiplication Based Logic Puzzle

Posts tagged ‘multiplication’

683 is the 4th Wagstaff Prime

683 is the sum of the five prime numbers from 127 to 149. Can you name those five prime numbers?

Like the number before it, 683 has a relationship with the number 11:

(2^11 + 1)/3 = 683. This relationship makes 683 the 4th Wagstaff Prime number. (Notice that 11 is the 4th odd prime number.)

2 raised to an odd prime number has produced many Wagstaff Prime numbers, but not always. For example (2^29 + 1)/3 is not a prime number.

683 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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  • 683 is a prime number.
  • Prime factorization: 683 is prime and cannot be factored.
  • The exponent of prime number 683 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 683 has exactly 2 factors.
  • Factors of 683: 1, 683
  • Factor pairs: 683 = 1 x 683
  • 683 has no square factors that allow its square root to be simplified. √683 ≈ 26.13427.

How do we know that 683 is a prime number? If 683 were not a prime number, then it would be divisible by at least one prime number less than or equal to √683 ≈ 26.1. Since 683 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 683 is a prime number.

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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.

683 Factors

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682 Deserves a Lot of Exclamation Points!!!

682 is the sum of the four prime numbers from 163 to 179. 682 is also the sum of the ten prime numbers from 47 to 89.

6 – 8 + 2 = 0 so 682 is divisible by 11.

Stetson.edu shared another amazing relationship between the number 682 and the number 11:

682 factorials

Besides the obvious inclusion of the digits 6-8-2, notice in the factorial expression that 11 is in each numerator and that 11 is also the sum of the numbers in each denominator.

682 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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  • 682 is a composite number.
  • Prime factorization: 682 = 2 x 11 x 31
  • 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 682 has exactly 8 factors.
  • Factors of 682: 1, 2, 11, 22, 31, 62, 341, 682
  • Factor pairs: 682 = 1 x 682, 2 x 341, 11 x 62, or 22 x 31
  • 682 has no square factors that allow its square root to be simplified. √682 ≈ 26.1151297.

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682 Factors

668 and Level 3

668 is the sum of consecutive prime numbers 331 and 337.

68 is divisible by 4 so 668 and every other number ending in 68 is divisible by 4.

80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 = 668

668 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

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  • The first ten multiples of 668 are 668, 1336, 2004, 2672, 3340, 4008, 4676, 5344, 6012, and 6680.
  • 668 is a composite number.
  • Prime factorization: 668 = 2 x 2 x 167, which can be written 668 = (2^2) x 167
  • 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 668 has exactly 6 factors.
  • Factors of 668: 1, 2, 4, 167, 334, 668
  • Factor pairs: 668 = 1 x 668, 2 x 334, or 4 x 167
  • Taking the factor pair with the largest square number factor, we get √668 = (√4)(√167) = 2√167 ≈ 25.845696.

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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.

668 Factors

662 More Candy Corn

662 is the sum of the twelve prime numbers from 31 to 79.

662 Puzzle Candy Corn

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-26

Here’s the same candy corn puzzle but less colorful.

662 Puzzle

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  • 662 is a composite number.
  • Prime factorization: 662 = 2 x 331
  • 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 662 has exactly 4 factors.
  • Factors of 662: 1, 2, 331, 662
  • Factor pairs: 662 = 1 x 662 or 2 x 331
  • 662 has no square factors that allow its square root to be simplified. √662 ≈ 25.72936.

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662 Logic

Ricardo tweeted the solution to the puzzle so I’m including it here as well.

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652 and Level 3

652 has 6 factors. 6 is a perfect number because all of its smaller factors, 1, 2, and 3, add up to its largest factor, 6.

The factors of 652 are 1, 2, 4, 163, and 326. The sum of those factors is 496, another perfect number. Note that all of 496’s smaller factors, 1, 2, 4, 8, 16, 31, 62, 124, and 248, add up to 496, its largest factor.

Stetson.edu states that 652 is the only known non-perfect number that produces a perfect number in both of those situations.

652 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 652 is a composite number.
  • Prime factorization: 652 = 2 x 2 x 163, which can be written 652 = (2^2) x 163
  • 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 652 has exactly 6 factors.
  • Factors of 652: 1, 2, 4, 163, 326, 652
  • Factor pairs: 652 = 1 x 652, 2 x 326, or 4 x 163
  • Taking the factor pair with the largest square number factor, we get √652 = (√4)(√163) = 2√163 ≈ 25.53429.

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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.

652 Factors

649 and Level 6

6 – 4 + 9 = 11 so 649 is divisible by 11.

649 is the short leg in exactly three Pythagorean triples. Can you determine which one is a primitive triple, and what are the greatest common factors of each of the two non-primitive triples?

  • 649-3540-3599
  • 649-19140-19151
  • 649-210600-210601

649 Puzzle

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

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  • 649 is a composite number.
  • Prime factorization: 649 = 11 x 59
  • 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 649 has exactly 4 factors.
  • Factors of 649: 1, 11, 59, 649
  • Factor pairs: 649 = 1 x 649 or 11 x 59
  • 649 has no square factors that allow its square root to be simplified. √649 ≈ 25.475478.

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649 Logic

648 and Level 5

648 is the sum of consecutive prime numbers 317 and 331.

The sixth root of 648 begins with 2.941682753. Notice all the digits from 1 to 9 appear somewhere in those nine decimal places. Stetson.edu states that 648 is the smallest number that can make that claim.

From Archimedes-lab.org I learned some powerful facts about the number 648:

  • 16² – 17² + 18² – 19² + 20² – 21² +22² – 23² + 24² – 25² + 26² – 27² + 28² – 29² + 30² – 31² + 32² = 648
  • 48² – 47² + 46² – 45² + 44² – 43² +42² – 41² + 40² – 39² + 38² – 37² + 36² – 35² + 34² – 33² = 648
  • (1^2)(2^3)(3^4) = 648
  • 18² + 18²  = 648
  • (6^3) + (6^3) + (6^3) =648

648 Puzzle

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

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  • 648 is a composite number.
  • Prime factorization: 648 = 2 x 2 x 2 x 3 x 3 x 3 x 3, which can be written 648 = (2^3) x (3^4)
  • The exponents in the prime factorization are 3 and 4. Adding one to each and multiplying we get (3 + 1)(4 + 1) = 4 x 5 = 20. Therefore 648 has exactly 20 factors.
  • Factors of 648: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, 648
  • Factor pairs: 648 = 1 x 648, 2 x 324, 3 x 216, 4 x 162, 6 x 108, 8 x 81, 9 x 72, 12 x 54, 18 x 36, or 24 x 27
  • Taking the factor pair with the largest square number factor, we get √648 = (√324)(√2) = 18√2 ≈ 25.455844122…

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648 Logic

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