Write an indirect proof.
Given:n^2 is even
Prove: n^2 is divisible by 4.
To write an indirect proof, also known as a proof by contradiction, we assume the opposite of what we want to prove, and then we derive a contradiction from that assumption.
Assumption: n^2 is not divisible by 4.
Since n^2 is not divisible by 4, it means it leaves a remainder when divided by 4. So, n^2 can be expressed as n^2 = 4k + r, where k is an integer and r is a non-zero remainder (r = 1, 2, or 3).
Now let's consider the given information that n^2 is even. If n^2 is even, then it can be written as n^2 = 2m, where m is an integer.
Substituting this value in the previously derived expression, we have:
2m = 4k + r
Rearranging the equation, we get:
2m - 4k = r
Factoring out 2 from the left side, we have:
2(m - 2k) = r
We can observe that the left side of the equation is a multiple of 2, which means r (the remainder) must also be even. However, we assumed earlier that r is a non-zero remainder, and the only even remainder possible is 0.
Thus, we have reached a contradiction: Assuming n^2 is not divisible by 4 leads to the conclusion that the remainder (r) must be 0, which contradicts our assumption.
Therefore, our initial assumption is false, and the indirect proof proves that n^2 is divisible by 4.