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 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
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Quote:
 Originally Posted by idontknow $\displaystyle 2a+3b =5ab$. Find all pairs (a,b) . $a= \dfrac{3b}{5b-2}$

For an integer $a$, $|3b| \ge |5b-2|$, also $|3b| = |5b-2| \times k, \; k \in \mathbb{Z}$

rest is easy....

Last edited by tahirimanov19; October 6th, 2019 at 05:37 AM. October 6th, 2019, 10:02 PM   #3
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Quote:
 Originally Posted by tahirimanov19 For an integer $a$, $|3b| \ge |5b-2|$
That's incorrect.

As $\displaystyle 0 < \frac{3b}{5b- 2} < 1$ for $b < 0$ or $b > 1$, $b = 0$ (so $a = 0$) or $b = 1$ (so $a = 1$). 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 $\displaystyle 5b-2$ 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. Tags integers, solve Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post idontknow Elementary Math 1 July 9th, 2019 01:36 AM idontknow Elementary Math 4 June 2nd, 2019 06:23 PM ultramegasuperhyper Number Theory 4 May 24th, 2011 09:19 PM Sara so Number Theory 3 November 12th, 2010 12:18 PM cafegurl Elementary Math 3 January 7th, 2009 07:21 AM

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