Lata wants to multiply a two-digit number with a three-digit number. When she simply placed the two-digit number to the left of the three-digit number, forming a five-digit number, it is found that this number is exactly nine times the product of the two numbers Lata took. Find the two numbers.

Let the two-digit number be $$x$$ and the three-digit number $$y$$; the five-digit number is $$1000x + y$$, so $$1000x + y = 9xy$$, which gives $$y = \frac{1000x}{9x - 1}$$. Since $$\gcd(x, 9x - 1) = 1$$, $$9x - 1$$ must divide 1000, and for $$10 \le x \le 99$$ it lies between 89 and 890. The divisors of 1000 in that range are 100, 125, 200, 250, 500, and only 125 leaves remainder 8 on division by 9 as $$9x - 1$$ must, so $$9x - 1 = 125$$ and $$x = 14$$. Then $$y = \frac{1000 \times 14}{125} = 112$$, and indeed $$14112 = 9 \times 14 \times 112$$. The numbers are 14 and 112.