A Multiplication Based Logic Puzzle

Archive for the ‘Puzzles’ 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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793 and Level 5

793 is the sum of two squares TWO different ways!

  • 28² + 3² = 793
  • 27² + 8² = 793

Notice that 793 is 4(198) + 1, and neither 28 and 3 or 27 and 8 have any common prime factors. Could 793 possibly be a prime number?

The answer is no for two reasons:

  1. √793 is about 28.2, so we only need to check to see if 793 is divisible by 5, 13, or 17 (all the 4N+1 prime numbers that are less than 28.) However, 793 ÷ 13 = 61, so  793 isn’t prime.
  2. Also, any number that can be written as the sum of two squares in more than one way is never a prime number.

Both 13 and 61 give a remainder of one when they are divided by four, and 793 is the hypotenuse of FOUR Pythagorean triples:

  • 143-780-793, which is 13 times 11-60-61
  • 168-775-793, a primitive calculated from 2(28)(3), 28² – 3², 28² + 3²
  • 305-732-793, which is 61 times 5-12-13
  • 432-665-793, a primitive calculated from 2(27)(8), 27² – 8², 27² + 8²

793 is also palindrome 191 in BASE 24; note that 1(24²) + 9(24) + 1(1) = 793

Finally Stetson.edu informs us that 793 is 2(397) – 1.

793-puzzle

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

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  • 793 is a composite number.
  • Prime factorization: 793 = 13 x 61
  • 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 793 has exactly 4 factors.
  • Factors of 793: 1, 13, 61, 793
  • Factor pairs: 793 = 1 x 793 or 13 x 61
  • 793 has no square factors that allow its square root to be simplified. √793 ≈ 28.16025568.

793-factor-pairs

791 and Level 4

To see if 791 is divisible by 7, you could try either one of these divisibility tricks:

  • 791 is divisible by 7 because 79 – 2(1) = 77 which obviously is divisible by 7.
  • 791 is divisible by 7 because 79 + 5(1) = 84 which most people know is 12 × 7.

791-puzzle

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

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  • 791 is a composite number.
  • Prime factorization: 791 = 7 x 113
  • 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 791 has exactly 4 factors.
  • Factors of 791: 1, 7, 113, 791
  • Factor pairs: 791 = 1 x 791 or 7 x 113
  • 791 has no square factors that allow its square root to be simplified. √791 ≈ 28.12472222.

791-factor-pairs

790 and Level 3

790’s prime factorization is 2 × 5 × 79. Since two of those numbers, 5 and 79, have a remainder of one when divided by four, I automatically know that 790 is the hypotenuse of exactly four Pythagorean triples.

790-puzzle

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

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  • 790 is a composite number.
  • Prime factorization: 790 = 2 x 5 x 79
  • 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 790 has exactly 8 factors.
  • Factors of 790: 1, 2, 5, 10, 79, 158, 395, 790
  • Factor pairs: 790 = 1 x 790, 2 x 395, 5 x 158, or 10 x 79
  • 790 has no square factors that allow its square root to be simplified. √790 ≈ 28.106939.

790-factor-pairs

789 and Level 2

789 consists of exactly three consecutive numbers so it is divisible by 3.

789-puzzle

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

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  • 789 is a composite number.
  • Prime factorization: 789 = 3 x 263
  • 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 789 has exactly 4 factors.
  • Factors of 789: 1, 3, 263, 789
  • Factor pairs: 789 = 1 x 789 or 3 x 263
  • 789 has no square factors that allow its square root to be simplified. √789 ≈ 28.08914.

789-factor-pairs

788 and Level 1

Since 88, its last two digits, are divisible by 4, we know that 788 and every other whole number ending in 88 is divisible by 4.

I learned the following fascinating fact about these six numbers starting with 788 from Stetson.edu:

788-consecutive-numbers

788 is also palindrome 404 in BASE 14. Note that 4(196) + 0(14) + 4(1) = 788.

788-puzzle

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

  • 788 is a composite number.
  • Prime factorization: 788 = 2 x 2 x 197, which can be written 788 = (2^2) x 197
  • 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 788 has exactly 6 factors.
  • Factors of 788: 1, 2, 4, 197, 394, 788
  • Factor pairs: 788 = 1 x 788, 2 x 394, or 4 x 197
  • Taking the factor pair with the largest square number factor, we get √788 = (√4)(√197) = 2√197 ≈ 28.071338.

788-factor-pairs

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

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