### Today’s Puzzle:

If you solve this Valentine’s multiplication table puzzle, you might just become love-struck with multiplication! It’s a level 6 puzzle so you might find most of the clues to be tricky. Be sure to use logic as you place each number from 1 to 10 in both the first column and also in the top row. There is only one solution.

### Factors of 1746:

- 1746 is a composite number.
- Prime factorization: 1746 = 2 × 3 × 3 × 97, which can be written 1746 = 2 × 3² × 97.
- 1746 has at least one exponent greater than 1 in its prime factorization so √1746 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1746 = (√9)(√194) = 3√194.
- The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1746 has exactly 12 factors.
- The factors of 1746 are outlined with their factor pair partners in the graphic below.

### More About the Number 1746:

Why is 1746 the sum of two squares? We can tell by looking at its PRIME factors:

2 is a power of 2,

3² is an even power, and

97 is a prime Pythagorean hypotenuse. (It is prime and one more than a positive multiple of 4).

At least one of its prime factors is a Pythagorean triple hypotenuse, and ALL of the rest of its non-Pythagorean-hypotenuse prime factors are either a power of 2 or combine to make a perfect square. Thus, 1746 is the sum of two squares. What are the two squares?

39² + 15² = 1746.

1746 is the hypotenuse of a Pythagorean triple:

1170-1296-1746, calculated from 2(39)(15), 39² – 15², 39² + 15².

It is also **18** times (65-72-**97**).