Question

How many two-digit numbers are divisible by 13?

Answer 1

Hint: As we know that the first two-digit number which is divisible by 13 is 13, the second one is 26 and so on but the last number is 91. Now we will use the concept of arithmetic progression to find the total number between 13 to 91.

Complete step-by-step answer:

We have the numbers which are divisible by 13 are as follows,
13, 26, 39.................91.

The above progression is an arithmetic progression having the first term is 13, the common difference is 13 and the last term is 91.

Now, we will find the total number of term in the arithmetic progression by the formula as given below;
$\text{last term=}a+(n-1)d$ where “a” is the first term , n is the total number of terms and “d” is the common difference of the arithmetic progression.

And we have a=13, n=? and d= 13,
\[\begin{align}
  & \Rightarrow 91=13+(n-1)13 \\
 & \Rightarrow 91-13=(n-1)13 \\
 & \Rightarrow \dfrac{78}{13}=(n-1) \\
 & \Rightarrow 6=(n-1) \\
 & \Rightarrow n=6+1=7 \\
\end{align}\]

Here, we can see that there are a total 7 terms in the above arithmetic progression.
Therefore, the total two digit number is 7 which is divisible by 13.

Note: Just remember the formula of the arithmetic progression for the last term very clearly and also go through the other formulae of arithmetic progression as it will be very helpful in these types of questions. Also be careful while doing calculation.