If you just want to be able to visualize the pattern, there's a wonderful little tool at http://www.cut-the-knot.org/Curriculum/ ... ngle.shtml
that shows you a pictorial representation of Pascal's Triangle modulo arbitrary numbers (e.g. a modulus of 3 diagram would have white circles where the number is divisible by 3).
There are certainly many obvious patterns (for example, the 3^Nth rows contain numbers that are all multiples of 3, other than the 2 1s on the ends, of course, and the 3^Nth-1 rows all have *no* numbers that are multiples of 3), but I'm not sure there's a simple equation that would define it for all rows. Based on some information in these sources, I suspect that it will tend to be on the order of the logarithm of the row number, but with wild fluctuations.
In the flavor of this forum, certainly the easiest way to find one of these values would be to write a little program that calculates it. There's a formula for calculating all the values of the Nth row of Pascal's Triangle in the wikipedia entry for it (for example), so you can even do it for an arbitrary row if you like, rather than generating all of them. It's possible that if you stare at that formula long enough you might come up with a way to represent this in a closed form.