If X and Y are tow different, positive, integer numbers .........?

Question

If X and Y are tow different, positive, integer numbers. and they can be : XY = 2(X+Y) then /X-Y/ is :
1 or 2 or 3 or 4 or 5 .
don't forget to explane your answer.

Answer 1

XY = 2(X + Y)

Expand RHS :
XY = 2X + 2Y

Subtract 2Y from both sides :
XY - 2Y = 2X

Factorise LHS :
Y(X - 2) = 2X

Divide both sides by (X - 2) :
Y = 2X / (X - 2)

Multiply both sides by -1 :
-Y = -2X / (X - 2)

Add X to both sides :
X - Y = X - 2X / (X - 2)

Compute RHS :
X - Y = [X(X - 2) - 2X] / (X - 2)

= (X^2 - 4X) / (X - 2)

By applying synthetic division, we get :
X - Y = (X - 2) - 4 / (X - 2)

Now, (X - 2) is always an integer.
But 4 / (X - 2) is only an integer for X = 1, 3, 4 or 6.
Thus, there are only 4 possibilities to try.

X = 1 gives X - Y = 3, so, |X - Y| = 3
X = 3 gives X - Y = -3, so, |X - Y| = 3
X = 4 gives X - Y = 0, so |X - Y| = 0
X = 6 gives X - Y = 3, so |X - Y| = 3

But the third one, where X - Y = 0, (implies that X = Y),
is not valid, because the question specifies that the
integers must be different.

Therefore, |X - Y| can only be equal to 3.

Answer 2

3

There are two possible solutions: (3,6) and (6,3), both of which satisfy the absolute value expression.

| 3 - 6 | = 3 and | 6 - 3 | = 3

For (6,3):
(6)(3)=2(6+3)
18=18

For (3,6):
(3)(6)=2(3+6)
18=18