+610 Find the ordered pair $(m,n),$ where $m,n$ are positive integers satisfying the following equation: $$6mn = 27 - 3m - 2n$$
+128460 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)
