### 409 Happy Birthday, John!

409 = (20^2) + (3^2) so it is the hypotenuse of a Pythagorean triple. That triple is [120, 391, 409]. Could 409 possibly be a prime number? The answer is at the end of the post.

Today is my son’s birthday. He lives on the other side of the country, so I’m making him a puzzle cake that he can devour. Many years ago when he still lived at home, he could solve any of my puzzles, so I know he can handle this one even though I haven’t revealed its difficulty level.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

I am grateful for facebook! His wife posted, “I felt like I was wrapping presents for a 10 year old boy this morning.” My son is a young father who loves, loves, loves Legos.

- 409 is a prime number.
- Prime factorization: 409 is prime and cannot be factored.
- The exponent of prime number 409 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 409 has exactly 2 factors.
- Factors of 409: 1, 409
- Factor pairs: 409 = 1 x 409
- 409 has no square factors that allow its square root to be simplified. √409 ≈ 20.2237

**How do we know that ****409**** is a prime number?** If 409 were not a prime number, then it would be divisible by at least one prime number less than or equal to √409 ≈ 20.2237. Since 409 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 409 is a prime number.

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