# If the product of two positive numbers is 3600, what is the minimum value of their sum?

Let the numbers be a and b. Given that the product of the numbers is 3600.

ab = 3600

We can use the AM-GM Inequality here to solve the problem. The arithmetic mean-geometric mean (AM-GM) inequality states that the arithmetic mean of positive real numbers is greater than or equal to the geometric mean of the numbers.

#### AM-GM Inequality

Let a and b are positive numbers, arithmetic mean is greater than or equal to geometric mean.

AM-GM Inequality states that:

or more in general for n numbers the equation states that :

Now to answer the question : we known the product ab = 3600.

replace the product of the two numbers in the equation

we get:

Square root of 3600 is 60. So we get

#### Answer:

Hence the minimum value of the sum of the numbers is 120.

Proof for the AM-GM equation for two numbers:

Square both sides, we get

Expand and rearrange the terms, we get:

This equation holds true, since square of a number is always positive i.e

Let y = a-b