October 6th, 2019, 03:57 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98  Solve for integers
$\displaystyle 2a+3b =5ab$. Find all pairs (a,b) . 
October 6th, 2019, 05:35 AM  #2 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 132 Thanks: 49 Math Focus: Area of Circle  $a= \dfrac{3b}{5b2}$ For an integer $a$, $3b \ge 5b2$, also $3b = 5b2 \times k, \; k \in \mathbb{Z}$ rest is easy.... Last edited by tahirimanov19; October 6th, 2019 at 05:37 AM. 
October 6th, 2019, 10:02 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,035 Thanks: 2271  
October 6th, 2019, 11:26 PM  #4 
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 
Just posted the equation to know when $\displaystyle 5b2 $ can divide $\displaystyle 3b$, which happens only when $\displaystyle b=1$. from now the value of $\displaystyle a$ is known. Also we can set 3b=0 , since it can be divided by all integers except 0. 

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