### Today’s Puzzle:

This weekend I laid a bouquet of red and white flowers on my husband’s grave and decided to make a red rose Memorial Day puzzle for the blog as well. It is a mystery-level puzzle.

Write the number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues. There is only one solution.

### Factors of 1765:

- 1765 is a composite number.
- Prime factorization: 1765 = 5 × 353.
- 1765 has no exponents greater than 1 in its prime factorization, so √1765 cannot be simplified.
- The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1765 has exactly 4 factors.
- The factors of 1765 are outlined with their factor pair partners in the graphic below.

### More About the Number 1765:

1765 is the sum of two squares in two different ways:

42² + 1² = 1765, and

33² + 26² = 1765.

1765 is the hypotenuse of FOUR Pythagorean triples:

84 1763 1765, calculated from 2(42)(1), 42² – 1², 42² + 1²,

413 1716 1765, calculated from 33² – 26², 2(33)(26), 33² + 26²,

1059-1412-1765, which is (3-4-**5**) times **353**, and

1125-1360-1765, which is **5** times (225-272-**353**).

1765 is a digitally powerful number:

1⁴ + 7³ + 6⁴ + 5³ = 1765.

1765 is a palindrome in a couple of different bases:

It’s A5A base 13 because 10(13²) + 5(13) + 10(1) = 1765, and

it’s 1D1 base 36 because 1(36²) + 13(36) + 1(1) = 1765.

How did you ever decide to explore a number in base 36?

I made a table. It goes to base 100, but I could easily make it go well beyond that if I wanted. I don’t always look very far down on the table when I write a post. When I wrote this one, I typed 1765 into the table, and it told me what that number was in several other bases.1765 is also 101 in base 42.