1280 and Level 6

To me, today’s level 6 puzzle looks a little like a puppy dog. If you know or use a multiplication table, then with proper training, finding the factors of this puzzle will be no problem.

Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

I’d like to tell you a little about the number 1280:

  • 1280 is a composite number.
  • Prime factorization: 1280 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5, which can be written 1280 = 2⁸ × 5
  • The exponents in the prime factorization are 8 and 1. Adding one to each and multiplying we get (8 + 1)(1 + 1) = 9 × 2 = 18. Therefore 1280 has exactly 18 factors.
  • Factors of 1280: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 640, 1280
  • Factor pairs: 1280 = 1 × 1280, 2 × 640, 4 × 320, 5 × 256, 8 × 160, 10 × 128, 16 × 80, 20 × 64, or 32 × 40
  • Taking the factor pair with the largest square number factor, we get √1280 = (√256)(√5) = 16√5 ≈ 35.77709.

1280 is the sum of the fourteen prime numbers from 61 to 127. Do you know what those prime numbers are?

32² + 16² = 1280

1280 is the hypotenuse of a Pythagorean triple:
768-1024-1280 which is (3-4-5) times 256
That triple can also be calculated from 32² – 16², 2(32)(16), 32² + 16²

Since 1280 is the 5th multiple of 256, I would expect that a number close to 1280 would be the 500th number whose square root could be simplified. That number was 1275, just five numbers ago.

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