# find (a,b)

• Apr 27th 2009, 11:36 AM
maria18
find (a,b)
Hello,

find all $(a,b)$ with $a,b \in \mathbb{N}^{*}$such that : $\displaystyle \frac {a}{b} + \frac {{21b}}{{25a}} \in\mathbb{N}$
• Apr 27th 2009, 12:18 PM
Opalg
Quote:

Originally Posted by maria18
Hello,

find all $(a,b)$ with $a,b \in \mathbb{N}^{*}$such that : $\displaystyle \frac {a}{b} + \frac {{21b}}{{25a}} \in\mathbb{N}$

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. (Evilgrin)
• Apr 28th 2009, 07:30 AM
maria18
Quote:

Originally Posted by Opalg
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. (Evilgrin)

you are wrong Opalg (Wait)
Spoiler:
$(a,b) = (3n,5n), n \in \mathbb{N}^{*}$ and $(a,b) = (7n,5n),n \in \mathbb{N}^{*}$ (Giggle)
• Apr 28th 2009, 07:41 AM
HallsofIvy
Maria18, how is your solution different from Opalg's?