You can also prove this with a direct algebraic proof:
Definiton: An integer n is said to be odd if it can be written as
n = 2k + 1
for some integer k.
Proof: Let n be the product of three consecutive odd numbers. Then,
`n = (2k + 1)(2k + 3)(2k + 5)`
`= 8k^3 + 36k^2 + 46k + 15`
`= 2(4k^3 + 18k^2 + 23k + 7) + 1`
Since integers are closed under addition and multiplication,
`4k^3 + 18k^2 + 23k + 7`
is an integer. Therefore n is odd.
Actually the product of any odd numbers is odd. I am not sure what kind of proof you are looking for. We can use the following argument, an even number is a number divisible by 2. Since all the prime factors of the product are actually the factors of the three odd numbers, 2 is not going to be one of the factors, in other words the product is not divisible by 2.
Another way to prove it will be by contradiction.
Instead of proving p=>q, we can prove that not q=> not p
Let n be the first odd integer, n+2 will be the secon, and n+4 will be the 3rd.
Suppose that p is an even number, then p is divisible by 2. Which means either n divisible by 2, or n+2 divisible by 2, or n+4 divisible by 2. The last statement is false, thus p is not even.