Exercise 31 Page 341

To prove a statement using an indirect proof, also known as proof by contradiction, we need to follow three steps.

1. Assume temporarily that the statement we want to prove is false and therefore its opposite is true.
2. Perform a logical reasoning until we reach a contradiction.
3. Conclude that the initial statement must be true since the assumption led to a contradiction.

We want to prove that an odd number is not divisible by $4.$ Therefore, we will start our proof by temporary assuming that an odd number is divisible by $4.$ Now let’s try to prove our assumption. Let $n$ be any integer number. We know that all even numbers are divisible by $2.$ Therefore, we know that $2n$ is an even number. We also know that adding $1$ to any even integer will give us an odd number. This allows us to write algebraic definitions for even numbers and odd numbers. If the odd number $2n+1$ is divisible by $4,$ then the quotient of $2n+1$ and $4$ is an integer. In other words the following fraction simplifies to be an integer. We will now simplify this fraction. Let's finally consider the obtained expression. If $n$ is even, then $0.5n$ is an integer number. The sum of an integer number and $0.25$ is not integer. Therefore, in this case we can conclude that $0.5n+0.25$ is not an integer number.

If $n$ is even, then $0.5n+0.25$ is not integer.

If $n$ is odd, then $0.5n$ is half an odd number, which is an not integer. Half an odd number plus $0.25$ is also not an integer. Therefore, for this case we can also conclude that $0.5n+0.25$ is not an integer number.

If $n$ is odd, then $0.5n+0.25$ is not integer.

We can see that there is no value of $n$ that for which the expression is an integer. This means that we reached a contradiction and the assumption must be false. Therefore, the initial statement must be true and an odd number is not divisible by $4.$