An engineer investigating mechanical stress determines that a critical component fails only when the stress factor $S=n^2+7n+5$ is a multiple of 3, where $n$ is an integer. The engineer wishes to prove that the component will never fail.

A proof by contradiction is initiated as follows:

'Assume there exists an integer $n$ such that $n^2+7n+5$ is divisible by 3.'

Case 1: $n=3k$ for some integer $k$.

$$S=(3k{)}^2+7(3k)+5=9k^2+21k+5=3(3k^2+7k+1)+2$$

As this expression leaves a remainder of 2 when divided by 3, it is not divisible by 3, which contradicts the initial assumption for this case.