A Multiplication Based Logic Puzzle

854

Stetson.edu informs us that you can find all the digits 1 to 9 exactly one time when 854 is combined with its square. I thought that was pretty cool so I made this gif:

854 and Its Square

make science GIFs like this at MakeaGif

854 can be written as the sum of consecutive numbers several ways:

  • as the sum of 4 consecutive numbers: 212 + 213 + 214 + 215 = 854
  • as the sum of the 7 consecutive numbers from 119 to 125 with 122 as the middle number.
  • as the sum of the 28 consecutive numbers from 17 to 44 with 30 and 31 as the middle numbers.
  • as the sum of 2 consecutive even numbers: 426 + 428 = 854
  • as the sum of the 7 consecutive even numbers from 116 to 128 with 122 as the middle number.
  • as the sum of the 14 consecutive even numbers from 48 to 74 with 60 and 62 as the middle numbers.
  • Even number 854 does not have any factor pairs in which both numbers are even, so it cannot be written as the sum of consecutive odd numbers.
854 is the hypotenuse of a Pythagorean triple: 154-840-854, which is 14 times (11-60-61).
  • 854 is a composite number.
  • Prime factorization: 854 = 2 × 7 × 61
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 854 has exactly 8 factors.
  • Factors of 854: 1, 2, 7, 14, 61, 122, 427, 854
  • Factor pairs: 854 = 1 × 854, 2 × 427, 7 × 122, or 14 × 61
  • 854 has no square factors that allow its square root to be simplified. √854 ≈ 29.223278.
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