Question

Which of the following numbers is divisible by $3$?
(a) \[18\]
(b) $60$
(c) $24$
(d) All of the above

Answer 1

Hint: As any number is divisible by $3$, when and only when the sum of digit of the numbers is completely divisible by $3$,
Like, $258$ here, sum of digit of number is $=\left( 2+5+8 \right)=15,$so, here sum of digit i.e. $15$ is completely divisible by $3$, hence, this number will be divisible by 3.
Apply this concept to determine the answer of the question.

Complete step-by-step solution:
As we have to determine that,
Among the numbers given in the options are divisible by $3$ or not.
As, any numbers are divisible by 3 only when the sum of digits of the given number is completely divisible by $3$.
So, applying this concept for finding the solution,
As, we have first number as $18$
So, sum of digit of $18$ will be $=\left( 1+8 \right)=9$
As, we know that, $9$ is completely divisible by $3$.
So, we can conclude that,
The given number $18$ will be divisible by $3$.
Now, considering the next number,
As, we have number as $60$
So, sum of digit of $60$ will be = $(6+0)$
As, we know that, $6$ is completely divisible by $3.$
So, we can conclude that,
The given number $60$ will be divisible by $3.$
Now, considering the next number,
As, we have number as $24$
So, sum of digit of $24$ will be $=\left( 2+4 \right)=6$
As, we know that, $6$ is completely divisible by $3.$
So, we can conclude that,
The given number $24$ will be divisible by $3.$
Hence,
As the all the numbers given in the options are divisible by $3.$

Option D is the correct answer.

Additional Information: As we know that,
Multiple of $3$ are $3,6,9,...$
So, we can conclude that,
If any number if divisible by $6$ or $9$ or by any other multiple of $3,$ then it will be also divisible by $3.$
Let us consider a number like $151,$
As here sum of digit of number is $=\left( 1+5+1 \right)=7$
As, we know that,
$7$ is not divisible by $3.$
So, the number $151$ will not be divisible by $3,$
But from here, we can also calculate the value of remainder when the given number is divisible by $3.$
For that, we need to divide the sum of digit i.e. $7$ by $3,$
As when we divide $7$ by $3,$ we will get $1$ as remainder,
So, when $151$ will be divisible by $3,$ it gives $1$ as a remainder.

Note: If a number is divisible by $3,$ it doesn’t mean that it will also be divisible by $3,$ but the vice-versa of statement is true, means if a number is divisible by any multiple of $3$ then it will be also divisible by $3.$
Like for a number, $276$
Here, sum of digit is $=\left( 2+7+6 \right)=15$
Here, we can also add the digit of obtained sum as sum of digit of sum i.e. $15$ will be $=\left( 1+5 \right)=6,$ so when we get the largest value of sum, then for simplifying it we can again sum the obtained sum of digit for checking whether it is divisible by $3$ or not.