A Multiplication Based Logic Puzzle

This is a level 1 puzzle that is easier than even most other level 1 puzzles. You can do this puzzle! If mathematics makes you uncomfortable, you can still do this puzzle! Even if math class is your worse nightmare, you can complete this puzzle, and gain a little confidence. Go ahead, give it a try! Figure out where each number from one to ten goes in the top row and also in the first column so that the puzzle turns into a mixed-up multiplication table. It’s easier and far less time consuming than Sudoku. You CAN do this puzzle! Then, after you find all the factors, and are feeling really good about yourself, IF you want, you can fill in all the other cells of this mixed up multiplication table.

Print the puzzles or type the solution on this excel file: 10-factors-853-863

853 is a prime number that leaves a remainder of 1 when divided by 4, so 853 is the hypotenuse of a Pythagorean triple: 205-828-853.

23² + 18² = 853 so 205-828-853 can be calculated from 23² – 18², 2(23)(18), 23² + 18².

  • 853 is a prime number.
  • Prime factorization: 853 is prime.
  • The exponent of prime number 853 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 853 has exactly 2 factors.
  • Factors of 853: 1, 853
  • Factor pairs: 853 = 1 × 853
  • 853 has no square factors that allow its square root to be simplified. √853 ≈ 29.20616

How do we know that 853 is a prime number? If 853 were not a prime number, then it would be divisible by at least one prime number less than or equal to √853 ≈ 29.2. Since 853 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 853 is a prime number.

Here’s another way we know that 853 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 23² + 18² = 853 with 23 and 18 having no common prime factors, 853 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √853 ≈ 29.2. Since 853 is not divisible by 5, 13, 17, or 29, we know that 853 is a prime number.



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