A Multiplication Based Logic Puzzle

677 and Level 4

677 = 26² + 1², and it is the hypotenuse of the primitive Pythagorean triple 52-675-677 which was calculated using 2(26)(1),  26² – 1²,  26² + 1².

677 is also the sum of the eleven prime numbers from 41 to 83.

677 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 677 is a prime number.
  • Prime factorization: 677 is prime and cannot be factored.
  • The exponent of prime number 677 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 677 has exactly 2 factors.
  • Factors of 677: 1, 677
  • Factor pairs: 677 = 1 x 677
  • 677 has no square factors that allow its square root to be simplified. √677 ≈ 26.01922.

How do we know that 677 is a prime number? If 677 were not a prime number, then it would be divisible by at least one prime number less than or equal to √677 ≈ 26.0. Since 677 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 677 is a prime number.

Here’s another way we know that 677 is a prime number: Since 677 ÷ 4 has a remainder of one, and 26² + 1² = 677, and 126 and 1 have no common prime factors, 677 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √677 ≈ 26.0. Since 667 is not divisible by 5, 13, or 17, we know that 677 is a prime number.

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677 Logic

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