First, we know that p = 501*503...597 and q = 501*503...597*599*601
Notice that p is completely contained within q, so we can rewrite: q = p*599*601
Ultimately, we want our answer to be in terms of q, so we can rearrange the above as: p = q/(599*601)
Now, let's substitute that into 1/p+1/q
We'll get 1/[q/(599*601)] + 1/q.
Dividing by a fraction is the same as multiplying by the reciprocal, so this becomes
599*601/q + 1/q
(599*601 +1)/q
At this point, you could just say we have some huge number over q and see the answer must be D, or we can do a little fun work with difference of squares to calculate precisely. 599*601 = (600-1)(600+1), so we're looking for
[(600-1)(600+1) + 1]/q
[600^2 - 1^2 + 1]/q
[360,000 -1 + 1]/q
360,000/q
The answer is D.
