Finding the Minimum Value of x + y for xy = 4225

The problem asks for the minimum value of the sum \(x + y\) given that \(x\) and \(y\) are natural numbers (positive integers) and their product \(xy\) is equal to 4225.

To find the minimum value of \(x + y\), we need to consider the pairs of natural numbers \(x\) and \(y\) whose product is 4225. These pairs are essentially the factor pairs of 4225.

Understanding Factors and the Product

When the product of two positive numbers is fixed, their sum is smallest when the numbers are as close to each other as possible. If the numbers can be equal, the minimum sum occurs when they are equal. If they cannot be equal, the minimum sum occurs when they are the pair of factors closest to the square root of the product.

In this case, the product is 4225. Let's find the square root of 4225: \[ \sqrt{4225} \] We can find this by prime factorization.

Prime Factorization of 4225

Let's break down 4225 into its prime factors:

• 4225 ends in 5, so it's divisible by 5. \(4225 \div 5 = 845\)
• 845 also ends in 5, so it's divisible by 5. \(845 \div 5 = 169\)
• 169 is \(13 \times 13\), which is \(13^2\).

So, the prime factorization of 4225 is \(5 \times 5 \times 13 \times 13 = 5^2 \times 13^2\).

The square root of 4225 is \(\sqrt{5^2 \times 13^2} = \sqrt{(5 \times 13)^2} = 5 \times 13 = 65\).

Since \(\sqrt{4225} = 65\), the minimum value of \(x + y\) will occur when \(x\) and \(y\) are equal to 65, provided that 65 is a natural number, which it is. And \(65 \times 65 = 4225\).

Listing Factors and Calculating Sums for xy = 4225

Let's list the pairs of natural numbers \(x\) and \(y\) such that \(xy = 4225\) and calculate their sums \(x + y\). The factors of \(4225 = 5^2 \times 13^2\) are \(5^a \times 13^b\) where \(0 \le a \le 2\) and \(0 \le b \le 2\).

(Note: Other pairs like (169, 25), (325, 13), etc., will just be the reverse, giving the same sum).

Identifying the Minimum Sum

Comparing the sums calculated: 4226, 850, 338, 194, and 130.

The minimum value among these sums is 130. This occurs when \(x = 65\) and \(y = 65\).

This confirms the principle that for a fixed product, the sum of two positive numbers is minimized when the numbers are equal or as close to each other as possible. Since \(4225 = 65 \times 65\), \(x=65\) and \(y=65\) is a valid pair of natural numbers satisfying \(xy = 4225\), and it gives the minimum sum.

Conclusion on Minimum Value

The minimum value of \(x + y\) when \(xy = 4225\) and \(x, y\) are natural numbers is 130.

Revision Table: Key Concepts for xy Product Problems

Additional Information: Sum and Product Relationship

The relationship between the sum and product of numbers is a fundamental concept in mathematics, often explored using inequalities. For two positive numbers \(x\) and \(y\), the Arithmetic Mean (AM) is \(\frac{x+y}{2}\) and the Geometric Mean (GM) is \(\sqrt{xy}\).

The AM-GM inequality states that for non-negative numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean: \[ \frac{x+y}{2} \ge \sqrt{xy} \] Equality holds if and only if \(x = y\).

If the product \(xy\) is fixed at \(P\), the inequality becomes: \[ \frac{x+y}{2} \ge \sqrt{P} \] \[ x+y \ge 2\sqrt{P} \]

This inequality shows that the minimum value of \(x + y\) is \(2\sqrt{P}\), and this minimum is achieved when \(x = y = \sqrt{P}\).

In our problem, \(P = 4225\). The minimum sum is \(2\sqrt{4225} = 2 \times 65 = 130\). This minimum is achieved when \(x = y = \sqrt{4225} = 65\). Since 65 is a natural number, this minimum value is attainable for natural numbers \(x\) and \(y\).

Understanding this relationship helps solve problems involving optimizing sums or products of numbers under certain conditions.