### Today’s Puzzle:

Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

### Factor Cake for 1664:

We can make a factor cake for 1664 by doing some successive divisions. Divide 1664 by 2, divide that answer by 2, and so forth until you make a factor cake that looks like this:

### Factors of 1664:

- 1664 is a composite number.
- Prime factorization: 1664 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 13, which can be written 1664 = 2⁷ × 13.
- 1664 has at least one exponent greater than 1 in its prime factorization so √1664 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1664 = (√64)(√26) = 8√26.
- The exponents in the prime factorization are 7 and 1. Adding one to each exponent and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 1664 has exactly 16 factors.
- The factors of 1664 are outlined with their factor pair partners in the graphic below.

### More About the Number 1664:

1664 is the sum of two squares:

40² + 8² = 1664.

That happened because it has a prime factor that leaves a remainder of 1 when divided by 4 AND all of its other prime factors are powers of 2 or perfect squares:

But that’s not all that cool about 1664. What patterns do you notice below?

2(24² + 16²) = 1664,

4(20² + 4²) = 1664,

8(12² + 8²) = 1664,

16(10² + 2²) = 1664,

32(6² + 4²) = 1664,

64(5² + 1²) = 1664, and

128(3² + 2²) = 1664.

1664 is the hypotenuse of a Pythagorean triple:

640-1536-1664, calculated from 2(40)(8), 40² – 8², 40² + 8².

That triple is also (5-12-**13**) times **128**.