1110 Another Mystery

The common factors of 36 and 12 are 1, 2, 3, 4, 6, and 12. If this were a Find the Factors 1-12 puzzle, you would have to consider most of those common factors. Since it’s a 1 to 10 puzzle, most of those factors aren’t allowed. Can you figure out the solution to this mystery puzzle?

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

Here are a few facts about the number 1110:

  • 1110 is a composite number.
  • Prime factorization: 1110 = 2 × 3 × 5 × 37
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1110 has exactly 16 factors.
  • Factors of 1110: 1, 2, 3, 5, 6, 10, 15, 30, 37, 74, 111, 185, 222, 370, 555, 1110
  • Factor pairs: 1110 = 1 × 1110, 2 × 555, 3 × 370, 5 × 222, 6 × 185, 10 × 111, 15 × 74, or 30 × 37
  • 1110 has no square factors that allow its square root to be simplified. √1110 ≈ 33.31666

1110 is the hypotenuse of FOUR Pythagorean triples:
342-1056-1110 which is 6 times (57-176-185)
360-1050-1110 which is 30 times (12-35-37)
624-918-1110 which is 6 times (104-153-185)
666-888-1110 which is (3-4-5) times 222

1110 looks very cool in some other bases:
It’s 5050 in BASE 6 because 5(6³) + 5(6) = 5(222) = 1110,
456 in BASE 16 because 4(16²) + 5(16) + 6(1) = 1110, and
UU in BASE 36 (U is 30 in base 10) because 30(36) + 30(1) = 30(37) = 1110

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