A Multiplication Based Logic Puzzle

960 Factor Trees

960 is the smallest number to have exactly 28 factors. 960 is 2⁶·3·5, so any factor tree made for it will have 6 + 1 + 1 = 8 prime factors. Since 8 is a power of 2, this number, 960, has some beautiful and well-balanced factor trees as well as some that aren’t as good-looking. Here are five of the MANY possible factor trees for 960:

 

960 can be written as the difference of 2 squares TEN different ways:

  1. 241² – 239² = (241 + 239)(241 – 239) = 480 × 2 = 960
  2. 122² – 118² = (122 + 118)(122 – 118) = 240 × 4 = 960
  3. 83² – 77² = (83 + 77)(83 – 77) = 160 × 6 = 960
  4. 64² – 56² = (64 + 56)(64 – 56) = 120 × 8 = 960
  5. 53² – 43² = (53 + 43)(53 – 43) = 96 × 10 = 960
  6. 46² – 34² = (46 + 34)(46 – 34) = 80 × 12 = 960
  7. 38² – 22² = (38 + 22)(38 – 22) = 60 × 16 = 960
  8. 34² – 14² = (34 + 14)(34 – 14) = 48 × 20 = 960
  9. 32² – 8² = (32 + 8)(32 – 8) = 40 × 24 = 960
  10. 31² – 1² = (31 + 1)(31 – 1) = 32 × 30 = 960

960 is the sum of the sixteen prime numbers from 29 to 97.

It is also the sum of six consecutive prime numbers:
149 + 151 + 157 + 163 + 167 + 173 = 960

960 is the hypotenuse of a Pythagorean triple:
576-768-960 which is (3-4-5) times 192

I like how 960 looks in these other bases:
33000 in BASE 4 because 3(4⁴) + 3(4³) = 3(256 + 64) = 3 × 320 = 960
440 in BASE 15 because 4(15²) + 4(15) = 4(225 + 15) = 4 × 240 = 960
UU in BASE 31 (U is 30 base 10), because 30(31) + 30(1) = 30(31 + 1) = 30 × 32 = 960
U0 in BASE 32 because 30(32) + 0 = 960

Stetson.edu informs us that 9 + 6 + 09³ + 6³ + 0³ = 960

  • 960 is a composite number and a perfect square.
  • Prime factorization: 960 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5, which can be written 960 = 2⁶ × 3 × 5
  • The exponents in the prime factorization are 6, 1 and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 960 has exactly 28 factors.
  • Factors of 960: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960
  • Factor pairs: 960 = 1 × 960, 2 × 480, 3 × 320, 4 × 240, 5 × 192, 6 × 160, 8 × 120, 10 × 96, 12 × 80, 15 × 64, 16 × 60, 20 × 48, 24 × 40, or 30 × 32
  • Taking the factor pair with the largest square number factor, we get √960 = (√64)(√15) = 8√15 ≈ 30.9838668

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