The LCM of all integers 2-9 is 2520. Then Y*(Y-1) = 2520*n, which implies Y = (1 +/- sqrt[1+10080*n])/2
The first n which makes Y an integer is n=20 yielding Y=225 and Y-1=224. But that has Y as a multiple of 3,5,9 and Y-1 as a multiple of 2,4,7,8 and neither a multiple of 6.
So I keep going until I find n=101 (the fifth n that yields integer Y) which yields Y=505 and Y-1=504. This almost works, Y is a multiple of 5 and Y-1 is a multiple of 2,3,4,6,7,8,9 - the exact opposite of what the problem asks.
Keep going until I finally find n=1612 (the eleventh n) which yields Y=2016 and Y-1=2015. This is what we want, Y is a multiple of 2,3,4,6,7,8,9 and Y-1 is a multiple of 5.
This is obviously slower than Steve's direct approach.