First, a remark about the shape of the proof. You are proving some statement (in this case, it is where is a positive integer). One can assume , then prove and thus obtain a contradiction with the assumption. Therefore, is false and is true. However, proof by contradiction is superfluous in your reasoning because in proving (the first time) you don't use the assumption . To put it simply, you prove directly, so why do you need at all?A1: Let n be any positive integer. Suppose does not hold. Thus, it is not the case that . Now, notice that . We can see that is even whenever is odd and vice-versa. Hence, either expression will at least be divisible by two, which is a factor of 6. Thus, which is the same as . This contradicts our original claim.
Next, you are right that one of and is even, so is also even. However, it does not follow from there that . For example, 4 is even, but 6 does not divide 4. You should factor further and conclude that one of the three factors of is divisible by 3.
By Bézout's theorem, is divisible by . You don't need to understand the proof; it is enough to perform the factorization.Prove or disprove: if is an integer greater than 2, is composite.