A Multiplication Based Logic Puzzle

Archive for the ‘Level 6 Puzzle’ Category

794 and Level 6

794 is the hypotenuse of a Pythagorean triple, 456-650-794, so 456² + 650² = 794².

794 is also palindrome 282 in BASE 18. Note that 2(18²) + 8(18) + 2(1) = 794.

Stetson.edu informs us that 1^6 + 2^6 + 3^6 = 1 + 64 + 729 = 794.

794-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 794 is a composite number.
  • Prime factorization: 794 = 2 x 397
  • 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 794 has exactly 4 factors.
  • Factors of 794: 1, 2, 397, 794
  • Factor pairs: 794 = 1 x 794 or 2 x 397
  • 794 has no square factors that allow its square root to be simplified. √794 ≈ 1780056.

794-factor-pairs

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787 Always a Unique Solution

  • 787 is a prime number.
  • Prime factorization: 787 is prime and cannot be factored.
  • The exponent of prime number 787 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 787 has exactly 2 factors.
  • Factors of 787: 1, 787
  • Factor pairs: 787 = 1 x 787
  • 787 has no square factors that allow its square root to be simplified. √787 ≈ 28.05352.

787-factor-pairs

Now for today’s puzzle….The fact that these Find the Factor puzzles always have a unique solution is an important clue in solving this rather difficult puzzle. Good luck!

787-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s more about the number 787:

787 is a palindrome in bases 4, 10, 11 and 16:

  • 30103 BASE 4; note that 3(256) + 0(64) + 1(16) + 0(4) + 3(1) = 787
  • 787 BASE 10; note that 7(100) + 8(10) + 7(1) = 787
  • 656 BASE 11; note that 6(121) + 5(11) + 6(1) = 787
  • 313 BASE 16; note that 3(256) + 1(16) + 3(1) = 787

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What did I mean when I wrote that the puzzles always having a unique solution is an important clue? There is only one clue in the puzzle that is divisible by 11. One of the rows and one of the columns do not have a clue, so the other 11 will go with one of them. The cell where the empty row and empty column intersect cannot be 132 because if that worked, it would produce two possible solutions to the puzzle. This table explains a logical order to find the solution.

787-logic

781 and Level 6

  • 781 is a composite number.
  • Prime factorization: 781 = 11 x 71
  • 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 781 has exactly 4 factors.
  • Factors of 781: 1, 11, 71, 781
  • Factor pairs: 781 = 1 x 781 or 11 x 71
  • 781 has no square factors that allow its square root to be simplified. √781 ≈ 27.94637722.

781-factor-pairs

Can you solve today’s puzzle?

781-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-2016

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Here’s more about the number 781:

781 is the sum of the 19 prime numbers from 7 to 79.

Thus 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 781

781 is a repdigit in base 5 and a palindrome in three other bases:

  • 11111 BASE 5. Note that 625 + 125 + 25 + 5 + 1 = 781.
  • 232 BASE 19. Note that 2(19²) + 3(19) + 2(1) = 781
  • 1J1 BASE 20, J = 19 base 10. Note that 1(20²) + 19(20) + 1(1) = 781
  • 141 BASE 26. Note that 1(26²) + 4(26) + 1(1) = 781

781 is also the sum of three squares five ways

  • 27² + 6² + 4² = 781
  • 24² + 14² + 3² = 781
  • 24² + 13² + 6² = 781
  • 21² + 18² + 4² = 781
  • 21² + 14² + 12² = 781

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781-logic

773 and Level 6

  • 773 is a prime number.
  • Prime factorization: 773 is prime and cannot be factored.
  • The exponent of prime number 773 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 773 has exactly 2 factors.
  • Factors of 773: 1, 773
  • Factor pairs: 773 = 1 x 773
  • 773 has no square factors that allow its square root to be simplified. √773 ≈ 27.8028775.

773-factor-pairs

How do we know that 773 is a prime number? If 773 were not a prime number, then it would be divisible by at least one prime number less than or equal to √773 ≈ 27.8. Since 773 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 773 is a prime number.

Here is today’s puzzle for you to try to solve:

773 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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What else is special about the number 773?

22² + 17² = 773 so 773 is the hypotenuse of the primitive Pythagorean triple 195-748-773 which was calculated using 22² – 17², 2(17)(22), 22² + 17².

Thus 195² + 748² + 773².

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

  • 26² + 9² + 4² = 773
  • 25² + 12² + 2² = 773
  • 24² + 14² + 1² = 773
  • 23² + 12² + 10² = 773
  • 22² + 15² + 8² = 773
  • 20² + 18² + 7² = 773

773 is a palindrome in two other bases:

  • 545 BASE 12, note that 5(144) + 4(12) + 5(1) = 773
  • 3D3 BASE 14 (D = 13 base 10); note that 3(196) + 13(14) + 3(1) = 773

Here’s another way we know that 773 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 22² + 17² = 773 with 22 and 17 having no common prime factors, 773 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √773 ≈ 27.8. Since 773 is not divisible by 5, 13, or 17, we know that 773 is a prime number.

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773 Logic

 


766 and Level 6

  • 766 is a composite number.
  • Prime factorization: 766 = 2 x 383
  • 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 766 has exactly 4 factors.
  • Factors of 766: 1, 2, 383, 766
  • Factor pairs: 766 = 1 x 766 or 2 x 383
  • 766 has no square factors that allow its square root to be simplified. √766 ≈ 27.6767.

766-factor-pairs

Here’s a level 6 puzzle for you to try:

766 Puzzle

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

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Here are a few more reasons to be interested in the number 766:

766 is the sum of the twelve prime numbers from 41 to 89.

766 is palindrome 23332 in BASE 4; note that 2(256) + 3(64) + 3(16) + 3(4) + 2(1) = 766.

766 is the sum of three squares six different ways:

  • 27² + 6² + 1² = 766
  • 26² + 9² + 3² = 766
  • 21² + 18² + 1² = 766
  • 21² + 17² + 6² = 766
  • 21² + 15² + 10² = 766
  • 19² + 18² + 9² = 766

766 is also the 18th centered pentagonal number because (5⋅17² + 5⋅17 + 2)/2 = 766.

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This table shows a way to solve puzzle #766 using logic:

766 Logic

759 and Level 6

  • 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

  • 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

 

 

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