Question

The no. of integral pairs(s) (x, y) whose sum is equal to their product is

Answer 1

Hint: To solve this question, first of all we have to take x and y as two numbers. It is given that the sum is equal to the product. That is, x + y = xy. Now, we have to rearrange the equation by taking all the ‘x’ terms to one side of the equal sign and ‘y’ terms on the other side. Then, substitute the integral values for ‘x’ and find the corresponding ‘y’ term.

Complete step-by-step answer:
Let us assume that x and y are two numbers.
It is given that the sum is equal to their products. So, we get a equation such as’
x + y = xy
Now, we have to rearrange the equation by taking all the ‘x’ terms to one side of the equal sign and ‘y’ terms on the other side. So the equation becomes,
x = xy – y
Now, as the y term is common in the right side of the equal sign, we can write the equation as,
x = y(x-1)
Now, take the x terms to one side.
\[\dfrac{x}{(x-1)}=y\]
As we are asked to find the no. of integral pairs where the sum is equal to their product, now we have to give values to x and find the value of y.
Case 1: When we take x = 0,
\[\dfrac{0}{(0-1)}=y\]
$\therefore y=0$.
So, the integral pairs will be (0, 0)
So, now we have to check whether the sum is equal to their products.
$0+0=0\times 0$
So, 0 = 0. Thus it satisfies the condition.
Case 2: When we take x = 1,
\[\dfrac{1}{(1-1)}=y\]
\[\Rightarrow \dfrac{1}{0}=y\]
So, y is not defined when x = 1.
Case 3: When we take x = 2,
\[\dfrac{2}{(2-1)}=y\]
\[\Rightarrow \dfrac{2}{1}=y\]
\[\therefore y=2\].
So, the integral pairs will be (2, 2)
So, now we have to check whether the sum is equal to their products.
$2+2=2\times 2$
So, 4 = 4. Thus it satisfies the condition.
Case 4: When we take x = 3,
\[\dfrac{3}{(3-1)}=y\]
\[\therefore y=\dfrac{3}{2}\]
Thus, y is not an integer. So, we cannot take $\left( 3,\dfrac{3}{2} \right)$ as the integral pair. But, this will satisfy the condition as,
$\begin{align}
  & 3+\dfrac{3}{2}=3\times \dfrac{3}{2} \\
 & \Rightarrow \dfrac{9}{2}=\dfrac{9}{2} \\
\end{align}$
But, we cannot take this into consideration because we are asked to find the integral pairs.
So, the no. of integral pairs(s) (x, y) ) whose sum is equal to their product is 2.
The integral pairs are (0, 0) and (2, 2).

Note: We have to remember that it is asked to find the integral pairs, so x and y must be integers. We have to check whether the condition is also satisfied. We can rearrange the equation either in terms of ‘x’ or ‘y’.