Ana multiplies two positive integers whose difference is 202, but makes a mistake and gets a number 1000 units smaller than the correct one. When she divides the result by the smaller of the numbers she should have multiplied, the quotient is 288 and the remainder is 67. What are the two numbers that Ana multiplied?

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Solution

Let $x$ and $x+202$ be the integers that Ana multiplies. From the result of Ana's division, we can write:

$$
x(x+202)-1000=288 x+67
$$

Thus,

$$
x^{2}-86 x-1067=0
$$

Solving the above equation, we find the roots to be $x=-11$ and $x=97$. Since $x$ is a positive integer, we must have $x=97$. The other number multiplied by Ana is $97+202=$ 299.

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