How many ordered pairs (x, y) exist which satisfy the following equation? Both x and y are whole numbers.

Question

$x \cdot\ y=2^2\cdot\ 3^4 \cdot\ 5^7 \cdot(x+y)$

I have tried rearranging the equation to different formats but not getting anywhere.

Answer 1

Simplifying this equation is really easy if you use a cool factoring trick.

$$xy=2^2 3^4 5^7(x+y)$$

Distribute the $2^2 3^4 5^7$.

$$xy=2^2 3^4 5^7x+2^2 3^4 5^7y$$

Subtract both sides by $2^2 3^4 5^7x+2^2 3^4 5^7y$.

$$xy-2^2 3^4 5^7x-2^2 3^4 5^7y=0$$

Now, add the product of the coefficients of $x$ and $y$, which is $(2^2 3^4 5^7)^2=2^4 3^8 5^{14}$, to both sides. (This helps us factor the left-hand side.)

$$xy-2^2 3^4 5^7x-2^2 3^4 5^7y+2^4 3^8 5^{14}=2^4 3^8 5^{14}$$

Factor the left-hand side.

$$(x-2^2 3^4 5^7)(y-2^2 3^4 5^7)=2^4 3^8 5^{14}$$

Now, we know $x-2^2 3^4 5^7$ is a factor of $2^4 3^8 5^{14}$, so we can just set $x-2^2 3^4 5^7$ to all of the positive and negative factors $2^4 3^8 5^{14}$, solve for $x$, substitute into the above equation, solve for $y$, and accept the solution if both $x$ and $y$ are non-negative. There are $675$ positive of $2^4 3^8 5^{14}$, so we get $675$ solutions overall from the positive factors.

We will now prove that the only solution from the negative factors is $(0, 0)$. We know $x$ must be non-negative. This means: $$x \geq 0$$ Subtract both sides by $2^3 3^4 5^7$. $$x-2^3 3^4 5^7 \geq -2^3 3^4 5^7$$ Take the absolute value of both sides. Remember to switch the inequality sign because right now, both sides are negative (the left-side is negative because we are only looking for solutions to negative factors right now) and by taking the absolute value, we are making it positive, meaning we are switching the sign. $$\lvert x-2^3 3^4 5^7 \rvert \leq 2^3 3^4 5^7$$

Since $2^3 3^4 5^7$ is the square root of $2^4 3^8 5^{14}$, the other factor must have an absolute value greater than or equal to $2^3 3^4 5^7$. Otherwise, the product will not be big enough to get to $2^4 3^8 5^{14}$. Thus, we get the following:

$$\lvert y-2^3 3^4 5^7 \rvert \geq 2^3 3^4 5^7$$

Now, $y-2^3 3^4 5^7$ must be negative since the other factor is negative, so we get:

$$y-2^3 3^4 5^7 \leq -2^3 3^4 5^7$$ Add both sides by $2^3 3^4 5^7$. $$y \leq 0$$

However, $y$ can not be negative, so $y=0$. Substitute back into the original equation ans solve for $x$ to get $x=0$. Therefore, the only solution involving negative factors is $(0, 0)$. If you include this solution, there are $676$ solutions. Otherwise, there are only $675$ solutions.