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Find the ordered pair $(m,n),$ where $m,n$ are positive integers satisfying the following equation: $$6mn = 27 - 3m - 2n$$

gueesstt  Apr 22, 2018
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6mn  = 27 - 3m - 2n     add 2n to both sides

 

6mn + 2n  =  27 - 3m      factor this

 

2n ( 3 + m)  =  3(9 - m)

 

Because of "2n"....the left side is always even

And the right side will be even when "m" is odd

Divide both sides by 3

 

(2/3)n (3 + m) = 9 - m

The right side is an integer

The left side will be an integer when "n" is a multiple of 3

 

Note that when n  = 3 and m  = 1, we have that

 

(2/3)(3)(3 + 1)  =  9 - 1

 

2(3 + 1)  =  9 - 1

 

2(4)  =  8       which is true

 

So....(m, n)  =  (1, 3) 

 

 

cool cool cool

CPhill  Apr 22, 2018

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