Question

How many 3-digit numbers are divisible by 6 in all?
(a)149
(b)150
(c)151
(d)166

Answer 1

Hint: To solve the question, we have to calculate the smallest 3-digit number divisible by 6 and the largest 3-digit number divisible by 6. To calculate the number of 3-digit numbers is divisible by 6, substitute the obtained smallest and largest 3-digit number value in the formula.

Complete step-by-step answer:
We know,
The smallest 3-digit number is 100.
The largest 3-digit number is 999.
The smallest 3-digit number divisible by 6 is 102.
The largest 3-digit number divisible by 6 is 996.
We know that the number of numbers between ‘a’ and ‘b’ divisible by n is equal to \[\left( \dfrac{a-b}{n} \right)+1\] such that a, b are multiples of n and a > b.
When compared with the obtained data, we get
The value of a = 996.
The value of b = 102.
The value of n = 6.
By substituting the values in the given equation, we get
The number of between 996 and 102 divisible by 6 is equal to
\[\begin{align}
  & \left( \dfrac{996-102}{6} \right)+1 \\
 & =\left( \dfrac{894}{6} \right)+1 \\
\end{align}\]
\[\begin{align}
  & =\left( \dfrac{6\times 149}{6} \right)+1 \\
 & =149+1 \\
 & =150 \\
\end{align}\]
Thus, the number of 3-digit numbers divisible by 6 is 150.
Hence, option (b) is the right choice.

Note: The possibility of mistake can be made using the general method by calculating all the possible numbers which are divisible by 6, which will increase the procedure of solving since the 3-digit multiples of 6 are large in number. The alternative way of solving can be by calculating the 3-digit numbers which are divisible by 2 and the 3-digit numbers which are divisible by 3. The numbers which are divisible by 2 and 3 are divisible by 6. Thus, we can arrive at the answer.