1020 A Week of Mystery

Sometimes changing things up a little is good. I decided to make a week’s worth of mystery level puzzles. The actual difficulty level will vary from puzzle to puzzle so give each one of them a try. If you think one is too easy or too difficult, the next one might not be. Here’s the first one:

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Here’s a little bit about the number 1020:

It is only 8 more than 1008, the previous number that also had 24 factors. Here are a few of its MANY possible factor trees.

1020 is the sum of six consecutive prime numbers:
157 + 163 + 167 + 173 + 179 + 181 = 1020

1020 is the hypotenuse of four Pythagorean triples:
156-1008-1020 which is 12 times (13-84-85)
432-924-1020 which is 12 times (36-77-85)
480-900-1020 which is (8-15-17) times 60
612-816-1020 which is (3-4-5) times 204

1020 looks interesting when it is written using some different bases:
It’s 33330 in BASE 4 because 3(4⁴ + 4³ + 4² + 4¹) = 3(340) = 1020,
848 in BASE 11 because 8(11²) + 4(11) + 8(1) = 1020,
606 in BASE 13 because 6(13²) + 6(1) = 6(170) = 1020,
480 in BASE 15 because 4(15²) + 8(15) = 4(225 + 30) = 4(255) = 1020,
390 in BASE 17 because 3(17²) + 9(17) = 3(289 + 51) = 3(340) = 1020,
UU in BASE 33 (U is 30 base 10) because 30(33) + 30(1) = 30(34) = 1020, and
U0 in BASE 34 because 30(34) = 1020

  • 1020 is a composite number.
  • Prime factorization: 1020 = 2 × 2 × 3 × 5 × 17, which can be written 1020 = 2² × 3 × 5 × 17
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1020 has exactly 24 factors.
  • Factors of 1020: 1, 2, 3, 4, 5, 6, 10, 12, 15, 17, 20, 30, 34, 51, 60, 68, 85, 102, 170, 204, 255, 340, 510, 1020
  • Factor pairs: 1020 = 1 × 1020, 2 × 510, 3 × 340, 4 × 255, 5 × 204, 6 × 170, 10 × 102, 12 × 85, 15 × 68, 17 × 60, 20 × 51, or 30 × 34
  • Taking the factor pair with the largest square number factor, we get √1020 = (√4)(√255) = 2√255 ≈ 31.9374

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