### Reducible Square Roots up to √765

- 765 is a composite number.
- Prime factorization: 765 = 3 x 3 x 5 x 17, which can be written 765 = (3^2) x 5 x 17
- The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 765 has exactly 12 factors.
- Factors of 765: 1, 3, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765
- Factor pairs: 765 = 1 x 765, 3 x 255, 5 x 153, 9 x 85, 15 x 51, or 17 x 45
- Taking the factor pair with the largest square number factor, we get √
**765**= (√9)(√85) = 3√85 ≈ 2**7**.**65**8633.

765 is the 300th number whose square root can be reduced! Here are three tables with 100 reducible square roots each showing all the reducible square roots up to √765. When three or more consecutive numbers have reducible square roots, I highlighted them.

That’s 300 reducible square roots found for the first 765 counting numbers. 300 ÷ 765 ≈ 0.392, so 39.2% of the numbers so far have reducible square roots.

Today’s puzzle is a whole lot less complicated than all that, so give it a try!

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

Logical steps to find the solution are in a table at the bottom of the post.

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Here are some other fun facts about the number 765:

765 is made from three consecutive numbers so it is divisible by 3. The middle of those numbers is 6 so 765 is also divisible by 9.

765 can be written as the sum of two squares two different ways:

- 27² + 6² = 765
- 21² + 18² = 765

Its other two prime factors, 5 and 17, have a remainder of 1 when divided by 4 so 765² can be written as the sum of two squares FOUR different ways, two of which contain other numbers that use the same digits as 765. Also notice that 9 is a factor of each number in the corresponding Pythagorean triples.

- 117² +
**756**² =**765**² - 324² + 693² = 765²
- 360² +
**675**² =**765**² - 459² + 612² = 765²

765 can also be written as the sum of three squares four different ways:

- 26² + 8² + 5² = 765
- 22² + 16² + 5² = 765
- 20² + 19² + 2² = 765
- 20² + 14² + 13² = 765

765 is a palindrome in two different bases:

- 1011111101 BASE 2; note that 1(512) + 0(256) + 1(128) + 1(64) + 1(32) + 1(16) + 1(8) + 1(4) + 0(2) + 1(1) = 765.
- 636 BASE 11; note that 6(121) + 3(11) + 6(1) = 765.

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