### 997 and Level 3

Can you fill in all the cells of this 12 × 12 mixed up multiplication table if all you are given are the clues given in this puzzle? I promise you it can be done. Start with the two clues at the top of the puzzle and work down clue by clue until you have found all the factors. Afterwards, filling in the rest of the multiplication table will be a breeze.

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

Here are a few facts about 997, the largest three-digit prime number:

31² + 6² = 997

That means 997 is the hypotenuse of a primitive Pythagorean triple:

372-925-997 calculated from 2(31)(6), 31² – 6², 31² + 6²

- 997 is a prime number.
- Prime factorization: 997 is prime.
- The exponent of prime number 997 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 997 has exactly 2 factors.
- Factors of 997: 1, 997
- Factor pairs: 997 = 1 × 997
- 997 has no square factors that allow its square root to be simplified. √997 ≈ 31.5753

**How do we know that 997 is a prime number?** If 997 were not a prime number, then it would be divisible by at least one prime number less than or equal to √997 ≈ 31.3. Since 997 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 997 is a prime number.

Here’s another way we know that 997 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 6² = 997 with 31 and 6 having no common prime factors, 997 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √997 ≈ 31.6. Since 997 is not divisible by 5, 13, 17, or 29, we know that 997 is a prime number.

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