The sum of two natural numbers is 8 and their product is 15

Let the required natural numbers be x and (8 − x).

It is given that the product of the two numbers is 15.

$\therefore x(8-x)=15$

$\Rightarrow 8 x-x^{2}=15$

$\Rightarrow x^{2}-8 x+15=0$

$\Rightarrow x^{2}-5 x-3 x+15=0$

$\Rightarrow x(x-5)-3(x-5)=0$

$\Rightarrow(x-5)(x-3)=0$

$\Rightarrow x-5=0$ or $x-3=0$

$\Rightarrow x=5$ or $x=3$

Hence, the required numbers are 3 and 5.