Question

find the no of integral solution (m,n)satisfying 2mn-5m+n=55

Answer 1

Given : 2mn-5m+n=55

To Find :  no of integral solution (m,n)satisfying

Solution:

2mn  -  5m  + n  = 55

=> 2mn + n  = 55 + 5m

=> n(2m + 1)  = 5 ( 11 + m)

=> n  = 5 ( m + 11)/(2m + 1)

Case 1  

2m + 1   = 5  => m  = 2

=> n  = 13

Case 2  

(2m + 1 ) k  =  m + 11

=>m(2k - 1)  = 11 - k  

=> m = (11 - k)/(2 k - 1)  

=> k = 1    ,  k = 2  ,   k = 4  

m = 10  ,  m = 3  ,  m  = 1  

m  = 1    => n  = 20

m  = 2 => n = 13

m = 3  =>  n = 10

m =  10  => n  = 5

4 Positive integral  solutions

also  ( if we consider non positive integral solutions as well )

2m + 1 = 1   => m = 0

m = 0    => n = 55

if we consider -ve integral solutions as well

2m + 1 = - 1  => m = -1   => n  = -50

m = (11 - k)/(2 k - 1)  

= > k = - 1  ,  k = - 3  ,  k  =  -10

=> m =  -4  ,  m = -2  ,  m = - 1

m = -2  => n = -15

m = - 4 => n = - 5

2m + 1   = -5

=> m = - 3

m = - 3 => n = - 8

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