Question

if a and b are distinct integers such that ab<1 and b ≠0, what is the greatest possible value of a/b

Answer 1

'a' and 'b' are integers so they are values drawn from the set {..., -3, -2, -1, 0, 1, 2, 3, ...} basically any positive or negative whole number, and we include 0 as well. If 'a' and 'b' are forced to be positive, then there are no solutions to a*b < 1. But if we include negative values, then a*b < 1 has solutions. For example (a,b) = (-2,1) is a solution because a*b = -2*1 = -2 which is less than 1.

It turns out that (a,b) = (0,k) which plugs into a/b = 0/k = 0 which is the (a,b) pairing that leads to the largest value of ab. We can use any number k as long as k is nonzero. If 'a' were nonzero, then 'a' would have to be negative while k is positive (or vice versa) to ensure that a*b is negative, but these results would make a/b smaller than 0, thus not the largest possible value of a/b

In short, the answer to your question is 0