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

Let the numbers be x and y.

Given,

Difference of two numbers is 7.

⇒ x - y = 7

⇒ x = 7 + y     ………(1)

Given,

Product of numbers is 450.

⇒ xy = 450     ………(2)

Substituting value of x from equation(1) in equation(2), we get :

⇒ (7 + y)y = 450

⇒ 7y + y² = 450

⇒ y² + 7y - 450 = 0

⇒ y² - 18y + 25y - 450 = 0

⇒ y(y - 18) + 25(y - 18) = 0

⇒ (y + 25)(y - 18) = 0

⇒ (y + 25) = 0 or (y - 18) = 0     [Using zero-product rule]

⇒ y = -25 or y = 18

y ≠ -25 [since they are natural numbers]

Substituting value of y = 18 in equation (1), we get :

⇒ x = 7 + 18 = 25

⇒ x = 25 and y = 18.

Hence, the two natural numbers are 18 and 25.