Question

The difference of two natural numbers is 7 and their product is 450. Find the numbers.

Answer 1

Let the smaller natural number be $$x$$.

Then, the larger natural number is $$(x + 7)$$.

According to the problem: $$x(x + 7) = 450$$

$$x^2 + 7x - 450 = 0$$ 

Splitting the middle term: $$x^2 + 25x - 18x - 450 = 0$$

$$x(x + 25) - 18(x + 25) = 0$$

$$(x + 25)(x - 18) = 0$$

$$x = -25 \quad \text{or} \quad x = 18$$

Since $$x$$ must be a natural number, reject $$x = -25$$.

Therefore, $$x = 18$$ and the other number is $$18 + 7 = 25$$.

Hence, the required numbers are 18 and 25.