Factorials versus triangle numbers
I was checking out the factorials versus triangle numbers and I ran across something that struck me as quite interesting.
Let's take 1! and divide it by the first triangle number, 1, which equals 1. Then I moved on to 2! and divided it by 3 (the second triangle number) which isn't evenly divided. Then I took 3! and divided it by 6 (the third triangle number) which equals 1. Then I took 4! and divided it by 10 which doesn't evenly divide. Next I tried dividing 5! by 15 which does evenly divide (equalling 8). Moving on to 6! and dividing it by 21 doesn't go evenly.
So far nothing surprising at this stage. However when I divided the next five factorials by their corresponding triangle numbers, there were no remainders! (the next factorial, 12!, does leave a remainder, but 13! doesn't and I stop here as this is as high as my calculator goes).
One would think that this interesting event would be on the internet somewheres or in a book, but I never ran across this before. This makes me wonder what the longest string would be by going into higher numbers (or at least what the longest string that's known).