### 795 a Level 1 Puzzle with ♥

I learned yesterday that pushing ALT on the computer key pad and 3 on the number pad at the same time produces a ♥ when the keys are released. I added a few ♥‘s to today’s puzzle because Valentine’s day is almost here:

- 795 is a composite number.
- Prime factorization: 795 = 3 x 5 x 53
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 795 has exactly 8 factors.
- Factors of 795: 1, 3, 5, 15, 53, 159, 265, 795
- Factor pairs: 795 = 1 x 795, 3 x 265, 5 x 159, or 15 x 53
- 795 has no square factors that allow its square root to be simplified. √795 ≈ 28.195744.

Stetson.edu informs us that if we add up all the factors of 795 we get a number that is a perfect 4th power:

- 1 + 3 + 5 + 15 + 53 + 159 + 265 + 795 = 1296, which happens to be the 4th power of 6. (Of course 1296 also equals 36 squared.)

795 is the hypotenuse of four Pythagorean triples and thus:

- 69 – 792 – 795 which is 3 times (23 – 264 – 265)
- 288 – 741 – 795 which is 3 times (96 – 247 – 265)
- 420 – 675 – 795 which is 15 times (28 – 45 – 53)
- 477 – 636 – 795 which is 159 times (3 – 4 – 5)

795 is also the sum of three squares two different ways:

- 25² + 13² + 1² = 795
- 25² + 11² + 7² = 795

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