Find two numbers whose sum is $27$ and product is $182$.

(N/A) Let the first number be $x$. Then the second number is $27-x$.
Given that their product is $182$,we have:
$x(27-x) = 182$
Expanding the equation:
$27x - x^2 = 182$
Rearranging into the standard quadratic form $ax^2 + bx + c = 0$:
$x^2 - 27x + 182 = 0$
Factoring the quadratic equation:
$x^2 - 13x - 14x + 182 = 0$
$x(x - 13) - 14(x - 13) = 0$
$(x - 13)(x - 14) = 0$
Setting each factor to zero:
$x - 13 = 0 \Rightarrow x = 13$
$x - 14 = 0 \Rightarrow x = 14$
If the first number is $13$,the second number is $27 - 13 = 14$.
If the first number is $14$,the second number is $27 - 14 = 13$.
Therefore,the two numbers are $13$ and $14$.