A number a is to be chosen at random from the set of {1,2,3,4,5,6}. A number b is to be chosen at random from the remaining five numbers. What is the probability that a/b will be an integer?
How can I prove that the answer is indeed 4/15?
I don't what to say that. With such a small outcome what would anyone want to do it the hard way?
Let $\displaystyle d_n$ be the number of divisors of $\displaystyle n$ less than $\displaystyle n$.
$\displaystyle d_1=0,~d_2=1,~d_3=1,~d_4=2,~d_5=1,~d_6=3$
There are eight such pairs out of thirty pairs.
If your set were $\displaystyle \{1,2,3,\cdots,98,99\}$ then the answer is: $\displaystyle \frac{{\sum\limits_{n = 2}^{99} {{d_n}} }}{{(99)(98)}}$
I am not sure if this simplifies the counting, but you can select b first and then a. This does not change the answer. For b = 1 there are 5 multiples of b among {1, 2, 3, 4, 5, 6} - {b}, for b = 2 there are 2 multiples, and for b = 3 there is 1 multiple. For other b's there are no multiples distinct from b in this set. The number of "successful" pairs (a, b) is 5 + 2 + 1 = 8.