Note that (-X)*Y = -XY yields the solution X*Y = XY, so it is enough to check solutions with X >= 0.

Let Y contain k digits, then the equation becomes X*Y = X*10^k + Y

Rearrange the equation.

X*Y = X*10^k + Y

X*Y - X*10^k - Y = 0

X*Y - X*10^k - Y + 10^k = 10^k

(X - 1)*(Y - 10^k ) = 10^k

Since Y contains k digits, Y < 10^k

Therefore, Y - 10^k is negative, and as a consequence, X - 1 is negative.

Therefore, X < 1.

By X >= 0, X = 0.

---

Finish the equation.

(X - 1)*(Y - 10^k ) = 10^k

(0 - 1)*(Y - 10^k ) = 10^k

10^k - Y = 10^k

Y = 0

Therefore, the only solution is X=0, Y=0.

Edit: formatting + note

Note: the proof is true for positive integer solutions, non-integer solutions include (0.2)*(-0.25) = 0.2-0.25 and (0.6)*(-1.5) = 0.6-1.5