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

Question

b*r=-d*s+1

Answer 1

If b=d=1 then r and s can be integers.

We can rewrite the equation:

s=(1-br)/d or r=(1-ds)/b.

So s=1/d-br/d or r=1/b-ds/b.

If d=1, s=1-br, so b must also be an integer.

If b=1, r=1-ds, so d must also be an integer.

If d=1/n, where n is an integer, s=n-nbr so b must also be an integer.

If b=1/m where m is an integer, r=m-mds so d must also be an integer.