Question

How many numbers in between 100 and 300 are divisible by 13?
A. 16
B. 15
C. 14
D. 13

Answer 1

Hint:We try to find the multiple numbers of 13 which remain between 100 and 300. We form them in the pattern of $13k,k\in \mathbb{Z}$ and try to find the values for $k$. The number of possible outcomes for $k$ gives us the solution.

Complete step by step answer:
The number divisible by 13 can be expressed as $13k,k\in \mathbb{Z}$.
We have to find the values of $k$ for which $100<13k<300$.
Now the highest multiple of 13 before 100 is 91 where $k$ is 7.
Therefore, $13\times 8=104$ is the first number in the domain of 100 to 300.
Now we go to find the maximum possible number before 300 and we divide it with 13.
$13\overset{23}{\overline{\left){\begin{align}
  & 300 \\
 & \underline{26} \\
 & 40 \\
 & \underline{39} \\
 & 1 \\
\end{align}}\right.}}$
So, we take the value of $k$ as 23.
We get $13\times 23=299$.
The possible values for $k$ is 8 to 23. There are in total 16 such values which means there are 16 such numbers in between 100 and 300 which are divisible by 13.

Hence, the correct answer is option A.

Note:We need to find the 3-digit lowest number divisible by 13 to find the starting point. We can also form the problem as the subtraction of the number of multiple under 100 from the number of multiple under 300. All the multiple numbers have to be natural numbers.