3. The product of two different natural numbers is 15 times greater than their sum. What values can the difference between the larger and smaller number take?
3. The product of two different natural numbers is 15 times greater than their sum. What values can the difference between the larger and the smaller number take?

## Solution.

Let the required numbers be $m$ and $n$. Assume $m > n$.

According to the condition of the problem, we have

$$
m n = 15(m + n)
$$

Now we get

$$
m n - 15 m - 15 n + 225 = 225
$$

or

$$
(m - 15)(n - 15) = 225
$$

Since $225 = 3^{2} \cdot 5^{2}$, and 225 is divisible by $n-15$ and $n < m$, we have the following possibilities:

$n-15=1, m-15=225$

$n-15=3, m-15=75$

$n-15=5, m-15=45$

$n-15=9, m-15=25$.

Then the possibilities for $m-n$ are as follows:

$$
m-n=224,72,40,16
$$

Remark. The equation $m n = 15(m + n)$ can be solved for $m$ to get

$$
m = \frac{15 n}{n - 15} = \frac{15(n - 15) + 225}{n - 15} = 15 + \frac{225}{n - 15}
$$

and then proceed in the same way as in the above solution.