Level 3 Puzzle

712 and Level 3

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  • 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

706 and Level 3

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  • 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

697 and Level 3

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  • 697 is a composite number.
  • Prime factorization: 697 = 17 x 41
  • 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 697 has exactly 4 factors.
  • Factors of 697: 1, 17, 41, 697
  • Factor pairs: 697 = 1 x 697 or 17 x 41
  • 697 has no square factors that allow its square root to be simplified. √697 ≈ 26.40075756.

Here is today’s factoring puzzle:

 

697 Puzzle

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

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Here are some more thoughts about the number 697:

Because both of its factors are hypotenuses of Pythagorean triples, 697 is the hypotenuse of FOUR Pythagorean triples:

  • 153-680-697
  • 185-672-697
  • 328-615-697
  • 455-528-697

Two of those triples are primitive, and two are not. Can you find the greatest common factor for each one that is not primitive?

697 is palindrome 151 in BASE 24; note that 24² = 576, and 1(576) + 5(24) + 1(1) = 697.

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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.

697 Factors

691 and Level 3

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  • 691 is a prime number.
  • Prime factorization: 691 is prime.
  • The exponent of prime number 691 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 691 has exactly 2 factors.
  • Factors of 691: 1, 691
  • Factor pairs: 691 = 1 x 691
  • 691 has no square factors that allow its square root to be simplified. √691 ≈ 26.2868789.

How do we know that 691 is a prime number? If 691 were not a prime number, then it would be divisible by at least one prime number less than or equal to √691 ≈ 26.3. Since 691 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 691 is a prime number.

Here is today’s puzzle:

691 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here’s a little more about 691:

691 is the sum of the thirteen prime numbers from 29 to 79.

691 is equivalent to palindrome 171 in BASE 23. Note that 1(23²) + 7(23) + 1(1) = 691.

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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.

691 Factors

683 is the 4th Wagstaff Prime

/ ivasallay / 3 Comments

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:

(2¹¹ + 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 (2²⁹ + 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.
  • 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

676 and Level 3

/ ivasallay / 2 Comments

676 is a palindrome in three consecutive different bases plus a couple of other bases:

  • 10201 in base 5; note that 1(625) + 0(125) + 2(25) + 0(5) + 1(1) = 676.
  • 676 in base 10; note that 6(100) + 7(10) + 6(1) = 676.
  • 565 in base 11; note that 5(121) + 6(11) + 5(1) = 676.
  • 484 in base 12; note that 4(144) + 8(12) + 4(1) = 676.
  • 121 in base 25; note that 1(625) + 2(25) + 1(1) = 676.

Speaking of palindromes, OEIS.org states that 676 is the smallest perfect square palindrome whose square root is not also a palindrome. (Palindromic perfect squares less than 676 are 1, 4, 9, 121, and 484.)

Since 13 and 169 are two of its factors, 676 is also the hypotenuse of Pythagorean triples 476-480-676 and 260-624-676. What is the greatest common factor of each triple?

676 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 676 is a composite number.
  • Prime factorization: 676 = 2 x 2 x 13 x 13, which can be written 676 = (2^2) x (13^2)
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 x 3 = 9. Therefore 676 has exactly 9 factors.
  • Factors of 676: 1, 2, 4, 13, 26, 52, 169, 338, 676
  • Factor pairs: 676 = 1 x 676, 2 x 338, 4 x 169, 13 x 52, or 26 x 26
  • 676 is a perfect square. √676 = 26

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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.

676 Factors

668 and Level 3

/ ivasallay / Leave a comment

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

661 Candy Corn

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25² + 6² = 661

661 is the hypotenuse of the primitive Pythagorean triple 300-589-661 which was calculated using 2(25)(6), 25² – 6², 25² + 6².

661 is also the sum of the three prime numbers from 211 to 227. What is the prime number in the middle of the sum?

This Find the Factors puzzle is supposed to look like a piece of candy corn. 🙂

661 Puzzle

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

Here’s the puzzle without the possibly distracting color:

661 Puzzle (plain)

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  • 661 is a prime number. 659 and 661 are twin primes.
  • Prime factorization: 661 is prime.
  • The exponent of prime number 661 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 661 has exactly 2 factors.
  • Factors of 661: 1, 661
  • Factor pairs: 661 = 1 x 661
  • 661 has no square factors that allow its square root to be simplified. √661 ≈ 25.70992.

How do we know that 661 is a prime number? If 661 were not a prime number, then it would be divisible by at least one prime number less than or equal to √661 ≈ 25.7. Since 661 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 661 is a prime number.

Here’s another way we know that 661 is a prime number: Since 25² + 6² = 661, and 25 and 6 have no common prime factors, 661 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √661 ≈ 25.7. Since 661 is not divisible by 5, 13, or 17, we know that 661 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 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.

661 Factors

652 and Level 3

/ ivasallay / Leave a comment

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.

OEIS.org 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

646 and Level 3

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646 is the hypotenuse of the Pythagorean triple 304-570-646. What 2-digit number is the greatest common factor of those three numbers?

646 Puzzle

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

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  • 646 is a composite number.
  • Prime factorization: 646 = 2 x 17 x 19
  • 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 646 has exactly 8 factors.
  • Factors of 646: 1, 2, 17, 19, 34, 38, 323, 646
  • Factor pairs: 646 = 1 x 646, 2 x 323, 17 x 38, or 19 x 34
  • 646 has no square factors that allow its square root to be simplified. √646 ≈ 25.41653.

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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.

646 Factors