Question

Given that the number 35a64 is divisible by 3, where a is a digit, what are the possible values of a?
A) 0
B) 3
C) 6
D) All of these

Answer 1

Hint: Here, we will be using the divisibility rule of 3 which states that “Sum of all the digits of the number, if divisible by 3 then that number will be divisible by 3.” So, we will here use this rule and will take the value of ‘a’ in multiple of 3 from 0-9 digits i.e. 0, 3, 6, 9.
Complete step-by-step answer:
In this question, we are asked to find what possible numbers are from 0-9 which we put in 35a64 and that whole number will be divisible by 3.
Here, we will use the divisibility rule of 3. This rule states that “Sum of all the digits of the number, if divisible by 3 then that number will be divisible by 3.”
Now, we are given number 35a64. Using this rule, we will add the digit present in number and see the total of the digit.
3+5+a+6+4 $=a+18$
Here, we know that 18 is multiple of 3 so, we have to select such values of which when added to 18 gives the value in multiple of 3.
So, selecting the digit from 0-9 and checking the values.
Taking digit $a=0$ here, so, we get $0+18=18$. So, 18 is divisible by 3 therefore, one value of a is 0.
Similarly, here we can think logically that as 18 is multiple of three then 21 comes in the table of 3 which will be divisible by 3. Here the value of a will be 3 as 18 + 3 will be 21.
Same way, after 21 comes 24 which is multiple of 3, so 18 + 6 will be 24. Thus, a will be 6.
Furthermore, 18 + 9 will be 27 divisible by 3. So, it will be 9.
Hence, values of a are 0, 3, 6, 9 which when kept in 35a64 will be divisible by 3.
Option (D) is correct.

Note: Students should be sure with the divisibility rule of 3 as, there are many such rules to be remembered. So, it might get confused and there are chances of mistakes. Another way of doing this is by taking the value of ‘a’ from 0-9 and checking the result by putting $a=1,2,3....9$. This will become very tedious and time consuming. So, it's better to go with the divisibility rule as it becomes easier to solve.