1160 Find the Factors Challenge

I love this particular puzzle. I had so much fun figuring out the logic. It’s a little bit complicated, but once you figure out the first move, it shouldn’t take too long to figure out most of the rest of the puzzle. Do give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-1148-1160

Here are some facts about the number 1160:

  • 1160 is a composite number.
  • Prime factorization: 1160 = 2 × 2 × 2 × 5 × 29, which can be written 1160 = 2³ × 5 × 29
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1160 has exactly 16 factors.
  • Factors of 1160: 1, 2, 4, 5, 8, 10, 20, 29, 40, 58, 116, 145, 232, 290, 580, 1160
  • Factor pairs: 1160 = 1 × 1160, 2 × 580, 4 × 290, 5 × 232, 8 × 145, 10 × 116, 20 × 58, or 29 × 40
  • Taking the factor pair with the largest square number factor, we get √1160 = (√4)(√290) = 2√290 ≈ 34.05877

26² + 22² = 1160
34² +  2² = 1160

1160 is the hypotenuse of FOUR Pythagorean triples:
136-1152-1160 which is 8 times (17-144-145)
(It can also be calculated from 2(34)(2), 34² –  2², 34² +  2²)
192-1144-1160 which is 8 times (24-143-145)
(It can also be calculated from 26² – 22², 2(26)(22), 26² + 22²)
696-928-1160 which is (3-4-5) times 232
800-840-1160 which is (20-21-29) times 40

1160 is a palindrome when it is written in these three bases:
It’s 808 in BASE 12 because 8(12²) + 8(1) = 8(145) = 1160,
525 in BASE 15 because 5(15²) + 2(15) + 5(1) = 1160, and
404 in BASE 17 because 4(17²) + 4(1) = 4(290) = 1160

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