Puzzle

649 and Level 6

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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

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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. OEIS.org 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

647 and Level 4

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647 is the sum of the five prime numbers from 113 to 139.

647 appears in only one Pythagorean triple, the primitive 647-209304-209305.

647 Puzzle

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

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  • 647 is a prime number.
  • Prime factorization: 647 is prime.
  • The exponent of prime number 647 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 647 has exactly 2 factors.
  • Factors of 647: 1, 647
  • Factor pairs: 647 = 1 x 647
  • 647 has no square factors that allow its square root to be simplified. √647 ≈ 25.43619.

How do we know that 647 is a prime number? If 647 were not a prime number, then it would be divisible by at least one prime number less than or equal to √647 ≈ 25.4. Since 647 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 647 is a prime number.

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647 Logic

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

644 Divisibility Rules and Level 2

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Divisibility rules 1-11 applied to the number 644:

  1. All counting numbers are divisible by 1 so 644 IS divisible by 1.
  2. 644 is even so it IS divisible by 2.
  3. 6 + 4 + 4 = 14 which is not a multiple of 3 so 644 is NOT divisible by 3.
  4. The last two digits, 44, are divisible by 4, so 644 IS divisible by 4.
  5. The last digit is not 0 or 5, so 644 is NOT divisible by 5.
  6. 644 is divisible by 2, but not by 3 so 644 is NOT divisible by 6.
  7. Breaking off the last digit, doubling it and subtracting it from the remaining digits we get: 64 – 2(4) = 56, a multiple of 7, so 644 IS divisible by 7.
  8. 6 is an even digit, and 44 is divisible by 4 but not by 8 so 644 is NOT divisible by 8.
  9. 6 + 4 + 4 = 14 which is not a multiple of 9 so 644 is NOT divisible by 9.
  10. The last digit is not 0, so 644 is NOT divisible by 10.
  11. 6 – 4 + 4 = 6 which is not a multiple of 11 so 644 is NOT a multiple of 11.

Just for the fun of it let’s try a divisibility rule for 23. Break off the last digit, multiply it by 7 and add it to the remaining digits: 64 + 7(4) = 64 + 28 = 92.

Now apply the same rule to 92: 9 + 7(2) = 9 + 14 = 23, obviously a multiple of 23, so 92 and 644 are both divisible by 23.

644 Puzzle

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

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  • 644 is a composite number.
  • Prime factorization: 644 = 2 x 2 x 7 x 23, which can be written 644 = (2^2) x 7 x 23
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 644 has exactly 12 factors.
  • Factors of 644: 1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644
  • Factor pairs: 644 = 1 x 644, 2 x 322, 4 x 161, 7 x 92, 14 x 46, or 23 x 28
  • Taking the factor pair with the largest square number factor, we get √644 = (√4)(√161) = 2√161 ≈ 25.377155

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644 Factors

643 and Level 1

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Here’s an interesting prime factorization: 2⁶⋅3⋅643 = 123456. Did you notice its largest prime factor? Thank you OEIS.org for that fun fact about the number 643.643 PuzzlePrint the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

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  • 643 is a prime number. 641 and 643 are twin primes.
  • Prime factorization: 643 is prime.
  • The exponent of prime number 643 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 643 has exactly 2 factors.
  • Factors of 643: 1, 643
  • Factor pairs: 643 = 1 x 643
  • 643 has no square factors that allow its square root to be simplified. √643 ≈ 25.357444666.

How do we know that 643 is a prime number? If 643 were not a prime number, then it would be divisible by at least one prime number less than or equal to √643 ≈ 25.4. Since 643 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 643 is a prime number.

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643 Factors

641 and Level 6

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25² + 4² = 641

641 is the hypotenuse of the primitive Pythagorean triple 200-609-641 which was calculated using 2(25)(4), 25² – 4², and 25² + 4² .

 641 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 641 is a prime number.
  • Prime factorization: 641 is prime.
  • The exponent of prime number 641 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 641 has exactly 2 factors.
  • Factors of 641: 1, 641
  • Factor pairs: 641 = 1 x 641
  • 641 has no square factors that allow its square root to be simplified. √641 ≈ 25.3179778.

How do we know that 641 is a prime number? If 641 were not a prime number, then it would be divisible by at least one prime number less than or equal to √641 ≈ 25.3. Since 641 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 641 is a prime number.

Here’s another way we know that 641 is a prime number: Since 25²+ 4² = 641, an odd number, and 25 and 4 have no common prime factors, we know that 641 is a prime number simply because it is not divisible by 5, 13, or 17.

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641 Logic

640 Fall Factor Trees and Level 5

/ ivasallay / 4 Comments

Because it is fall, and 640 has many factors, I decided to make factor trees using fall colors. Get out your rakes!

640 factor trees

There are many other possible factor trees for 640, but raking leaves can be a lot of work, so I only made two of them.

640 is the hypotenuse of the Pythagorean triple 384-512-640.

OEIS.org informs us that 640 = 16!!!!!!, but if you type 16!!!!!! into a calculator, you will get an error message as soon as you type !!.

16!!!!!! ≠ (((((16!)!)!)!)!)!

There are 6 !’s so 16!!!!!! = 16(16-6)(16-12) = 16 x 10 x 4.

Here is this week’s Level 5 puzzle:

640 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 640 is a composite number.
  • Prime factorization: 640 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 640 = (2^7) x 5
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 640 has exactly 16 factors.
  • Factors of 640: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640
  • Factor pairs: 640 = 1 x 640, 2 x 320, 4 x 160, 5 x 128, 8 x 80, 10 x 64, 16 x 40, or 20 x 32
  • Taking the factor pair with the largest square number factor, we get √640 = (√64)(√10) = 8√10 ≈ 25.298221

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640 Logic

639 and Level 4

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639 is the sum of the 20 prime numbers from 2 to 71, its largest prime factor.

639 is made from 3 numbers that are divisible by 3 so 639 is divisible by 3 AND by 9.

Puzzle #639 has a tricky clue in it, but I’m sure you can still solve it.

639 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 639 is a composite number.
  • Prime factorization: 639 = 3 x 3 x 71, which can be written 639 = (3^2) x 71
  • 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 639 has exactly 6 factors.
  • Factors of 639: 1, 3, 9, 71, 213, 639
  • Factor pairs: 639 = 1 x 639, 3 x 213, or 9 x 71
  • Taking the factor pair with the largest square number factor, we get √639 = (√9)(√71) = 3√71 ≈ 25.278449.

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639 Logic

638 and Level 3

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638 is the sum of the four prime numbers from 151 to 167.

6 – 3 + 8 = 11. Thus 638 is divisible by 11.

638 is also the hypotenuse of the Pythagorean triple 440-462-638. What is the greatest common factor of those three numbers?

638 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 638 is a composite number.
  • Prime factorization: 638 = 2 x 11 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 638 has exactly 8 factors.
  • Factors of 638: 1, 2, 11, 22, 29, 58, 319, 638
  • Factor pairs: 638 = 1 x 638, 2 x 319, 11 x 58, or 22 x 29
  • 638 has no square factors that allow its square root to be simplified. √638 ≈ 25.25866.

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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.

638 Factors