### Mysterious 1023

2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ + 2² + 2¹ + 2⁰ = 1023. That makes 1023 a pretty cool and rather mysterious number.

This puzzle that I’ve numbered 1023 is pretty cool and mysterious, too. I’m sure you will enjoy solving it if you only use logic to find the solution.

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

Here are some other fascinating facts about the number 1023:

It is formed by using a zero and three other consecutive numbers, so it is divisible by 3.

1 – 0 + 2 – 3 = 0, so 1023 is divisible by eleven.

31 × 33 = 1023 so (32 – 1)(32 + 1) = 1023, AND it is 32² – 1, making it one away from the next square number!

It is the sum of five consecutive prime numbers:

193 + 197 + 199 + 211 + 223 = 1023

1023 looks quite interesting when it is written in several different bases:

First of all, it’s 1111111111 in BASE 2 because it is the sum of the all those powers of 2 from 0 to 9 that were included at the top of this post.

It’s also 33333 in BASE 4 because 3(4⁴ + 4³ + 4² + 4¹ + 4⁰) = 3(341) = 1023.

That also means that 3(2⁸ + 2⁶ + 2⁴ + 2² + 2⁰) = 1023

It’s 393 in BASE 17 because 3(17²) + 9(17) + 3(1) = 1023,

VV in BASE 32 (V is 31 base 10) because 31(32) + 31(1) = 31(33) = 1023, and

V0 in BASE 33 because 31(33) = 1023

- 1023 is a composite number.
- Prime factorization: 1023 = 3 × 11 × 31
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1023 has exactly 8 factors.
- Factors of 1023: 1, 3, 11, 31, 33, 93, 341, 1023
- Factor pairs: 1023 = 1 × 1023, 3 × 341, 11 × 93, or 31 × 33
- 1023 has no square factors that allow its square root to be simplified. √1023 ≈ 31.98437

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