1321 Christmas Gift Puzzle

 

This puzzle looks like a gift because it is! You can fill out an entire multiplication table just from the 16 clues in this puzzle.

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Here are a few facts about the number 1321:

  • 1321 is a prime number.
  • Prime factorization: 1321 is prime.
  • The exponent of prime number 1321 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1321 has exactly 2 factors.
  • Factors of 1321: 1, 1321
  • Factor pairs: 1321 = 1 × 1321
  • 1321 has no square factors that allow its square root to be simplified. √1321 ≈ 36.34556

How do we know that 1321 is a prime number? If 1321 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1321 ≈ 36.3. Since 1321 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1321 is a prime number.

 

1321 is the sum of nine consecutive prime numbers:
127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 = 1321

1321 is the sum of two squares:
36² +  5² = 1321

1321 is the hypotenuse of a primitive Pythagorean triple:
360-1271-1321 calculated from 2(36)(5), 36² –  5², 36² +  5²

Here’s another way we know that 1321 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 36² + 5² = 1321 with 36 and 5 having no common prime factors, 1321 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1321 ≈ 36.3. Since 1321 is not divisible by 5, 13, 17, or 29, we know that 1321 is a prime number.

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