Question

When a number is divided by 13, the remainder is 11. When the same number is divided by 17, the remainder is 9. What is the number?
A.339
B.349
C.369
D.Data inadequate

Answer 1

Hint: We can form 2 equations from the given details for the required number by using 2 variables as the quotients. Then we can equate the 2 equations. Then we can find the values of and such that both are natural numbers. So, we substitute for one variable and check whether the other variable becomes a natural number. Thus, we can obtain the values of the variables. Substituting them back will give the required number.

Complete step-by-step answer:
Let N be the required number. Let ${q_1}$ be the quotient when divided by 13. It is given that we get a remainder of 11 when divided by 13. We can write it as an equation.
   $N = 13 \times {q_1} + 11$ … (1)
Let ${q_2}$ be the quotient when divided by 17. It is given that we get a remainder of 9 when divided by 17. We can write it as an equation.
 $N = 17 \times {q_2} + 9$ … (2)
Now we can equate (1) and (2),
 $13 \times {q_1} + 11 = 17 \times {q_2} + 9$
On simplification, we get,
 $ \Rightarrow 13 \times {q_1} = 17 \times {q_2} - 2$
On solving for ${q_1}$ , we get
 $ \Rightarrow {q_1} = \dfrac{{17 \times {q_2} - 2}}{{13}}$
We need to find the values of ${q_1}$ and ${q_2}$ such that both are natural numbers. So, we can substitute for different values ${q_2}$ such that ${q_1}$ is also a natural number. To get a idea of how big ${q_2}$ is , we can divide on of the given option a by 17.
 $ \Rightarrow a = \dfrac{{339}}{{17}}$
On division we get,
 $ \Rightarrow a = 19.94$
So, we can check the equation for ${q_2} = 20$
 $ \Rightarrow {q_1} = \dfrac{{17 \times 20 - 2}}{{13}}$
On simplification of numerator we get,
 $ \Rightarrow {q_1} = \dfrac{{340 - 2}}{{13}}$
 $ \Rightarrow {q_1} = \dfrac{{338}}{{13}}$
On division we get,
 $ \Rightarrow {q_1} = 26$
Thus ${q_1}$ is a natural number for ${q_2} = 20$ .
Now we can substitute, ${q_2} = 20$ is equation (2), to get the required number.
 $ \Rightarrow N = 17 \times {q_2} + 9$
 $ \Rightarrow N = 17 \times 20 + 9$
 $ \Rightarrow N = 340 + 9$
 $ \Rightarrow N = 349$
So, the required number is 349.
So the correct answer is option B.


Note: We used the concept of division algorithm to write the two equations. According to the division algorithm, for every number m and n, there exists unique r and q such that $n = mq + r$where q is called the quotient, r is the remainder and m is the divisor. After finding the equation, we use the method of trial and error to find the values of ${q_1}$ and ${q_2}$ .
Alternate method to solve this problem is given by,
Let N be the required number. Let ${q_1}$ be the quotient when divided by 13. It is given that we get a remainder of 11 when divided by 13. We can write it as an equation.
 $N = 13 \times {q_1} + 11$
On rearranging, we get,
 $ \Rightarrow 13 \times {q_1} = N - 11$
 \[ \Rightarrow {q_1} = \dfrac{{N - 11}}{{13}}\]
Now we can substitute for N for the values given in the option.
When $N = 339$
 \[ \Rightarrow {q_1} = \dfrac{{339 - 11}}{{13}}\]
 \[ \Rightarrow {q_1} = \dfrac{{328}}{{13}}\]
328 is not divisible by 13. So, $N = 339$ is false.
Now we can check $N = 349$
 \[ \Rightarrow {q_1} = \dfrac{{349 - 11}}{{13}}\]
 \[ \Rightarrow {q_1} = \dfrac{{338}}{{13}}\]
 \[ \Rightarrow {q_1} = 26\]
Therefore, $N = 349$ is the required number.