1036 Look, Look to the Rainbow

Finian’s Rainbow is a wonderful movie to enjoy on Saint Patrick’s Day. One of its songs reminds us to “Look, look to the rainbow”.

If you look to this rainbow, you will find all the factors of 1036:

There is a simple symmetry in every rainbow. There is also symmetry in palindromes which are numbers, words, or sentences that read the same forward or backward.

1036 demonstrates that symmetry when it is written in some other bases:
It’s repdigit 4444 in BASE 6 because 4(6³ + 6² + 6¹ + 6⁰) = 4(216 + 36 + 6 + 1) = 4(259) = 1036,
It’s 232 in BASE 22 because 2(22²) + 3(22) + 2(1) = 1036,
1M1 in BASE 23 (M is 22 base 10) because 23² + 22(23) + 1 = 1036, and
SS in BASE 36 (S is 28 base 10) because 28(36 + 1) = 28(37) = 1036

1036 is also the hypotenuse of a Pythagorean triple:
336-980-1036 which is 28 times 12-35-37

  • 1036 is a composite number.
  • Prime factorization: 1036 = 2 × 2 × 7 × 37, which can be written 1036 = 2² × 7 × 37
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1036 has exactly 12 factors.
  • Factors of 1036: 1, 2, 4, 7, 14, 28, 37, 74, 148, 259, 518, 1036
  • Factor pairs: 1036 = 1 × 1036, 2 × 518, 4 × 259, 7 × 148, 14 × 74, or 28 × 37,
  • Taking the factor pair with the largest square number factor, we get √1036 = (√4)(√259) = 2√259 ≈ 32.18695

There wasn’t a pot of gold at the end of our factor rainbow, but there is one here at the end of this post. It’s a level 5 puzzle, but it isn’t too difficult, so see if you can find all the factors that make the puzzle function like a multiplication table.

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043

 

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