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I forgot to do that part of #3, so here it is:
The formula was n(n+1)/2, where n is an integer. If n is even, then we can rearrange the formula to (n/2)(n+1), and both factors must be integers. If n is odd, we know that n+1 must be even, so (n)((n+1)/2) has two integer factors. Since n is an integer, it has to be even or odd, and this formula always gives an integer result.