If the sum of two numbers is divided by $15$, the quotient is $2$ and the remainder is $10$. If the difference of the same numbers is divided by $3$ then the quotient is $4$ and the remainder is $2$. Find the numbers.

Hint: In this problem, we need to find two numbers. To find the required numbers, we will use a division algorithm. We know that the division algorithm states that Dividend $ = $ (Divisor $ \times $ Quotient) $ + $ Remainder. After applying the algorithm, we will use a simple elimination method to solve the two equations.


Complete step-by-step answer:
We know that the division algorithm states that Dividend $ = $ (Divisor $ \times $ Quotient) $ + $ Remainder.
Let $x$ and $y$ be two required numbers. In this problem, it is given that the sum of two numbers, $x$ and $y$ here, when divided by $15$, the quotient is $2$ and the remainder is $10$. From this information, we can say that the dividend is $x + y$ and the divisor is $15$. Therefore, we can write $x + y = \left( {15 \times 2} \right) + 10$. Let us simplify this equation. We will get $x + y = 40 \cdots \cdots \left( 1 \right)$.
Also it is given that if the difference of the same numbers, $x$ and $y$ here, is divided by $3$ then the quotient is $4$ and the remainder is $2$. From this information, we can say that the dividend is $x - y$ and the divisor is $3$. Therefore, we can write $x - y = \left( {3 \times 4} \right) + 2$. Let us simplify this equation. We will get $x - y = 14 \cdots \cdots \left( 2 \right)$.
Now we have the following equations:
$x + y = 40 \cdots \cdots \left( 1 \right)$
$x - y = 14 \cdots \cdots \left( 2 \right)$
We need to find values of $x$ and $y$ from these two equations. For this, we will use the simple elimination method. If we add equations $\left( 1 \right)$ and $\left( 2 \right)$ then $y$ will be eliminated and we will get the value of $x$. Let us add equations $\left( 1 \right)$ and $\left( 2 \right)$. Therefore, we get
$
  \left( {x + y} \right) + \left( {x - y} \right) = 40 + 14 \\
   \Rightarrow x + y + x - y = 54 \\
   \Rightarrow 2x = 54 \\
   \Rightarrow x = \dfrac{{54}}{2} \\
   \Rightarrow x = 27 \\
 $
To find the value of $y$, we will substitute the value $x = 27$ in either equation $\left( 1 \right)$ or $\left( 2 \right)$. Let us substitute $x = 27$ in equation $\left( 1 \right)$. Therefore, we get
$
  27 + y = 40 \\
   \Rightarrow y = 40 - 27 \\
   \Rightarrow y = 13 \\
 $
Therefore, the required numbers are $27$ and $13$.


Note: The division algorithm is also applicable when we are dealing with polynomials. If we know one factor (divisor) of the polynomial, then we can find other factors by using the division algorithm.