find all $\displaystyle (a,b)$ with $\displaystyle a,b \in \mathbb{N}^{*} $such that :$\displaystyle \displaystyle \frac {a}{b} + \frac {{21b}}{{25a}} \in\mathbb{N}$
Clearly if (a,b) is a solution then so is (ca,cb) for any natural number c. So the only interesting solutions will be those for which a and b have no common divisor. There are two such solutions.
Spoiler:
They are (a,b) = (3,5) and (a,b) = (7,5). Now prove that there are no others.
Clearly if (a,b) is a solution then so is (ca,cb) for any natural number c. So the only interesting solutions will be those for which a and b have no common divisor. There are two such solutions.
Spoiler:
They are (a,b) = (3,5) and (a,b) = (7,5). Now prove that there are no others.
you are wrong Opalg
Spoiler:
$\displaystyle (a,b) = (3n,5n), n \in \mathbb{N}^{*}$ and $\displaystyle (a,b) = (7n,5n),n \in \mathbb{N}^{*}$