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

Let first number be x.

Since the difference of two numbers is 7, so other number is x + 7.

Given, the products of numbers = 450

⇒ x(x + 7) = 450

⇒ x² + 7x = 450

⇒ x² + 7x - 450 = 0

⇒ x² + 25x - 18x - 450 = 0

⇒ x(x + 25) - 18(x + 25) = 0

⇒ (x + 25)(x - 18) = 0

⇒ x + 25 = 0 or x - 18 = 0

⇒ x = -25 or x = 18.

Since, the numbers are natural numbers,

∴ x ≠ -25

∴ x = 18, x + 7 = 25

Hence, the numbers are 18 and 25.