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(r+1)! = (r+1) * r!
So lcm(r!, (r+1)!) divides (r+1)!
We can add the fractions on the LHS using the denominator (r+1)!
You should quickly see why it works out.
By the way, LHS means left hand side of the equation, and if you didn't understand the line about lcm, it's not terribly important, as long as you can see why (r+1)! can be used as a common denominator.
Edit:
Actually, the stronger statement
lcm(r!, (r+1)!) = (r+1)!
holds. I just wrote the weaker statement because I knew it was safe, without having to think. But obviously lcm(a, b) >= max(a, b) so it was a pretty silly precaution.