If the product of two successive positive integers is $7482$, which is the greatest integer?

Let the two consecutive positive integers be $x$ and $x+1$.
According to the problem description:

$$x(x+1)=7482$$

$$x^2+x-7482=0$$

To find the roots of this quadratic equation, notice that $\sqrt{7482}$ lies near $86$ because $80^2=6400$ and $90^2=8100$.
Let's test numbers:

$$86\times 87=7482$$

Hence, the two successive positive integers are $86$ and $87$. The larger integer among them is $87$.