What conditions must a,b,d satisfy so that r and s can both be integers?

Question

(a/b)*r-(a/(b*d))=s

Answer 1

The equation can be written:

(a/b)(r-(1/d))=s.

If r is an integer then 1/d must also be an integer, so d=1 or d is a fraction in which 1 is the numerator, such as ½, ¼, ⅕, ⅓, etc. The other condition is that b is a factor of a (when a and b are integers) so that a/b is an integer. These two conditions will create integer s when r is an integer. a and b can also be fractions provided a/b is an integer: e.g., a=1½ and b=½.

If a=b=d=1, then r and s are consecutive integers.