Appreciating even and odd primes

One of the nice problem which makes you appreciate is I like to call Farm
House problem, it goes something like this, There is a farm where there are different prime numbers of cows, goats and rabbits and if we add the number of cows and goats and multiply that by the number of cows we get 120 plus the number of rabbits. Determine the number of ech of these animals.
To work this out start with the smallest prime number 2 as the number of rabbit and the right side becomes 122 and it can be factorize as 2*(2+59), which is a contradiction as this will imply that there are same number of cows and rabbits. Then you might want to check if cows could be 2 and that would imply that
2*(2+g)=120+r, however this imply that the right side is even and the left side is odd (why ?), therefore cows cannot be 2. Let's check if the goat could be 2 and this time our equation takes the form c*(c+2)=120+g. The left side is odd as we are multiplying 2 odd numbers and the right side is also odd as we are adding a odd and a even number. Which means the number of goats could be 2. The question arises is that really true or it could be some other number prime number too? Let's assume that its some other prime then the left side of equation becomes even as the sum of two odd numbers is even and the right side of the equation is odd as sun of odd and even is odd and we get the parity contradiction. This we are now absolutely sure that the number of goats is 2 and then it's only s matter of plug and chug to discover that number if cows will be 11 and number of goats is 23.