April 6th, 2014, 01:05 PM  #1 
Senior Member Joined: Nov 2013 Posts: 434 Thanks: 8  m and n
2^m .5^n +2^3+5^n=101 find m+n 
April 6th, 2014, 01:36 PM  #2 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
Do you mean $2^m5^n+2^\color{red}m+5^n=101$?

April 6th, 2014, 05:09 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  
April 14th, 2014, 05:16 AM  #4 
Senior Member Joined: Jul 2013 From: Croatia Posts: 180 Thanks: 11 
If he really meant $\displaystyle 2^m \cdot 5^n+2^m+5^n=101$ $\displaystyle 2^m=x $ $\displaystyle 5^n=y $ $\displaystyle xy+x+y=101 $ $\displaystyle xy+x+y+1=102 $ $\displaystyle (x+1)(y+1)=102=51 \cdot 2=34 \cdot 3 =17 \cdot 6$ ; x+1 is always odd and y+1 is always even the only integer solution is when $\displaystyle 2^m+1=17 \Rightarrow m=4 $ $\displaystyle 5^n+1=6 \Rightarrow n=1 $ m+n=5 