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