## A Multiplication Based Logic Puzzle

### 963 Arrow

The numbers 1 to 12 fit someplace in the first column as well as in the top row.  Can you figure out where those places are so that this puzzle can become a multiplication table?

Print the puzzles or type the solution in this excel file: 12 factors 959-967

Now I’d like to mention a few things about the number 963:

It is the sum of the 24 prime numbers from 2 to 89. Pretty cool!

I like the way 963 looks in base 10 as well as in a few other bases:
33003 in BASE 4 because 3(4⁴) + 3(4³) + 3(1) = 3 × 321 = 963
3C3 in BASE 16 (C is 12 in base 10) because 3(16²) + 12(16) + 3(1) = 963
1B1 in BASE 26 (B is 11 in base 10) because 1(26²) + 11(26) + 1(1) = 963
123 in BASE 30 because 1(30²) + 2(30¹) + 3(30⁰) = 963

• 963 is a composite number.
• Prime factorization: 963 = 3 × 3 × 107, which can be written 963 = 3² × 107
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 963 has exactly 6 factors.
• Factors of 963: 1, 3, 9, 107, 321, 963
• Factor pairs: 963 = 1 × 963, 3 × 321, or 9 × 107
• Taking the factor pair with the largest square number factor, we get √963 = (√9)(√107) = 3√107 ≈ 31.03224