Finding Two Positive Integers

Let the two positive integers be denoted by $x$ and $y$. We are given two conditions about these integers. Let's assume $x$ is the larger integer and $y$ is the smaller integer without losing generality, meaning $x > y$.

The first condition states that the sum of the squares of these two positive integers is 306. We can write this as an equation:

$\text{Equation 1: } x^2 + y^2 = 306$

The second condition states that the square of the larger integer ($x$) is 25 times the smaller integer ($y$). We can write this as another equation:

$\text{Equation 2: } x^2 = 25y$

Solving the System of Equations

Now we have a system of two equations with two variables, $x$ and $y$. We can use substitution to solve this system. Since Equation 2 gives us an expression for $x^2$, we can substitute this expression into Equation 1.

Substitute $x^2 = 25y$ into $x^2 + y^2 = 306$:

$25y + y^2 = 306$

Rearrange the terms to form a standard quadratic equation:

$y^2 + 25y - 306 = 0$

Solving the Quadratic Equation for the Smaller Integer

We need to solve the quadratic equation $y^2 + 25y - 306 = 0$ for $y$. We can try to factor the quadratic expression. We are looking for two numbers that multiply to -306 and add up to +25.

Let's list factors of 306 and see if their difference is 25:

• 1 and 306 (Difference = 305)
• 2 and 153 (Difference = 151)
• 3 and 102 (Difference = 99)
• 6 and 51 (Difference = 45)
• 9 and 34 (Difference = 25)

The numbers 34 and 9 have a difference of 25. Since the middle term is $+25y$ and the constant term is $-306$, the factors must be $(y + 34)$ and $(y - 9)$.

So, the factored equation is:

$(y + 34)(y - 9) = 0$

This gives us two possible values for $y$:

• $y + 34 = 0 \implies y = -34$
• $y - 9 = 0 \implies y = 9$

The problem states that the integers are positive. Therefore, $y$ must be a positive integer. We discard the solution $y = -34$.

So, the smaller positive integer is $y = 9$.

Finding the Larger Integer

Now that we have the value of $y$, we can find the value of $x$ using Equation 2:

$x^2 = 25y$

Substitute $y=9$ into this equation:

$x^2 = 25 \times 9$

$x^2 = 225$

To find $x$, we take the square root of 225. Since $x$ is a positive integer, we take the positive square root:

$x = \sqrt{225} = 15$

So, the larger positive integer is $x = 15$.

Verifying the Integers

Let's check if these two integers, 15 and 9, satisfy the original conditions:

• Are they positive integers? Yes, 15 and 9 are positive integers.
• Is the sum of their squares 306? $15^2 + 9^2 = 225 + 81 = 306$. Yes, this condition is met.
• Is the square of the larger integer (15) 25 times the smaller integer (9)? $15^2 = 225$. $25 \times 9 = 225$. Yes, this condition is met.

The two positive integers are 15 and 9.

Calculating the Difference Between the Two Integers

The question asks for the difference between the two integers. We assumed $x$ is the larger integer and $y$ is the smaller integer, so the difference is $x - y$.

Difference = $15 - 9 = 6$

The difference between the two positive integers is 6.