We’re in FOM, we should be able to do a simple proof, right?!
Okay, so let’s try and prove this:
If n is a positive integer such that is an integer, then n equals 1.
Well, first of all, the beauty of “if, then statements” is that we can already assume half of this is true right off the bat! For this statement to be true, we’ll assume the hypothesis is true. That is, we’re assuming that n must be a positive integer, and that is an integer. Now all we have to prove is that n can only equal 1 with all other conditions already true! Easy peasy!
Let’s break this down a little further into cases.
Case 1: n > 1. If n is greater than 1, it fulfills the first part of the hypothesis, as it is positive (greater than one), and that it is a whole number (integer), but when put under a one, this awesome integer becomes a big bad fraction, which does not fulfill the second part of the hypothesis, that has to be an integer. 1 over any number bigger than 1 is a fraction, which is not an integer. Throw ‘em out.
Case 2: 0 < n < 1. Alright. If n is between 0 and 1, either becomes an integer (like with 1/2), which fulfills the second part of the hypothesis (yay!) or doesn’t (like with 2/5). Either way, these numbers in this range are positive, but not whole numbers, and therefore have to be thrown out because they do not fulfill the first part of the hypothesis! Throw ‘em out.
Case 3: n < 0. This does not fulfill the first part of the hypothesis! n must be positive. Throw ‘em out.
Case 4: n = 1. AH. So 1 is a positive, whole number. Which fulfills the first part of the hypothesis. And when 1 is divided by itself, it becomes 1 (surprise), which is an integer! Therefore, we can keep one!
Through these 4 cases, I’ve hopefully shown all possible outcomes of different integer types to prove that only n works!
Which leads me to ask, did I just explain the proof over again, or did I actually prove this statement right?