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

gueesstt
Apr 22, 2018

#1**+1 **

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)

CPhill
Apr 22, 2018