1104 and Level 3

If this were a Find the Factors 1-12 puzzle, the possible common factors for 12 and 48 would be 4, 6, and 12. But we can only have factors from 1 to 10 so only one of those common factors will work with this puzzle. If you know which one, you are well on your way to solving it.

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here are some facts about the number 1104:

  • 1104 is a composite number.
  • Prime factorization: 1104 = 2 × 2 × 2 × 2 × 3 × 23, which can be written 1104 = 2⁴ × 3 × 23
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1104 has exactly 20 factors.
  • Factors of 1104: 1, 2, 3, 4, 6, 8, 12, 16, 23, 24, 46, 48, 69, 92, 138, 184, 276, 368, 552, 1104
  • Factor pairs: 1104 = 1 × 1104, 2 × 552, 3 × 368, 4 × 276, 6 × 184, 8 × 138, 12 × 92, 16 × 69, 23 × 48, or 24 × 46
  • Taking the factor pair with the largest square number factor, we get √1104 = (√16)(√69) = 4√69 ≈ 33.2265.

1104 is the sum of the sixteen prime numbers from 37 to 103. Do you know what those prime numbers are?

1104 is also the sum of eight consecutive primes and two consecutive primes:
113 + 127 + 131 + 137 + 139 + 149 +151 + 157  = 1104
547 + 557 = 1104

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