Question

How many numbers between 1 and 300 are divisible by 3 and 5?
(a) 10
(b) 19
(c) 25
(d) 30

Answer 1

Hint: In the above problem, we are asked to find the number of numbers between 1 and 300 which are divisible by both 3 and 5. The numbers lying between 1 and 300 which are divisible by both 3 and 5 are the numbers which are divisible by 15 so basically, we need to find the numbers between 1 and 300 which are divisible by 15.

Complete step-by-step solution
To find the total number of numbers lying between 1 and 300 which are divisible by 3 and 5 we are going to find the numbers lying between 1 and 300 and are divisible by 15.
We have to exclude 1 and 300 in finding the total of numbers which are divisible by 15.
The numbers which are lying between 1 and 300 are divisible by 15 are as follows:
15, 30, 45, 60………285
As you can see that the above series is in A.P. (or Arithmetic Progression) in which the first term is 15 and the common difference is calculated by taking any term and subtracting this term from its succeeding term.
Let us take any term from the above series to find the common difference of the above A.P. say 45. Now, the number which is the successor of 45 is 60 so subtracting 45 from 60 we get,
$\begin{array}{rl}& 60-45\\ & =15\end{array}$
Hence, the common difference of the above A.P. is 15.
We know that the general term of A.P. is equal to:
$T_n=a+(n-1)d$ …………….. Eq. (1)
In the above formula, $T_n$ is the general term of an A.P., “a” is the first term, “d” is a common difference, and “n” is the order of the term whether the term is first, second or third.
Substituting $T_n=285$, “a” as 15 and “d” as 15 in eq. (1) we get,
$\begin{array}{rl}& 285=15+(n-1)15\\ & \Rightarrow 285=15+15n-15\\ & \Rightarrow 285=15n\end{array}$Dividing 15 on both the sides we get,
$\begin{array}{rl}& {\displaystyle \frac{285}{15}}=n\\ & \Rightarrow 19=n\end{array}$
From the above calculations, we have got the number of terms of the above A.P. is 19 and this number 19 is the number of terms lying between 1 and 300 and are divisible by 3 and 5.
Hence, the correct option is (b).

Note: Instead of using the formula for A.P. in counting the number of terms between 1 and 300 which are divisible by 15 you can write all the numbers which are divisible by 15 starting from 15 till 285 and then count them and come to a conclusion that there are 19 terms.
The advantage of using the formula for the general term of A.P. is that it will save your time in exams and you can do more questions.