Solution below.

Reveal solution by highlighting:

The expression (n!-123)/(n-123) is an integer, when n-123 is a factor of n!-123.

123 is interesting, because it’s part of the expression. 123 by itself wouldn’t work, because the denominator would be 0. What about n = 2*123 = 246? Here the denominator is 123, and this would work, because n > 123, so 123 is a factor of n!, and of course 123/123 = 1. How about n = 247? Here the denominator becomes 124. So 247!-123 must have 124 as a factor. So 247!+1 also has 124 as a factor. But 247! already has 124 as a factor, so that won’t work. Something similar happens for every possible n thereafter. So, n = 246.