Question

# State whether True or False:If a number is divisible by $9$, it must be divisible by $3$A. TrueB. False

Hint: Here we use the concept that any number $a$ when divided by $b$ can be written as $a = bq + r$ where $q$ is the quotient and $r$ is the remainder. A number is completely divisible by any number if its remainder is zero i.e. it can be written in the form $a = bq$. So we write number divisible by $9$ then further factoring nine we write if the number is divisible by factors of $9$

General equation of a number $a$ completely divisible by $b$ is $a = bq$.
Since, we are given a number is divisible by $9$, so put the value $b = 9$ in the above equation.
We can write $a = 9q$
Since, we can write nine in simpler form i.e. $9 = 3 \times 3$
Therefore, we can write $a = 9q = (3 \times 3)q$
Group together all the factors other than $3$
$a = 3 \times (3q)$
Assuming the factor $3q = p$
We can write $a = 3p$
which is of the form $a = bq$, where $b = 3$
Therefore, number $a$ is divisible by $3$
Since, the number on the LHS of the equation is the same, we can say the number that is divisible by $9$ is also divisible by $3$ .
So, the statement in the question is true.

So, the correct answer is “Option A”.

Note: Students many times make mistake when they assume looking at the word divisible by a number and they write the number in fraction form which is wrong, keep in mind that whenever a number is divisible by another number then the first number can be written as a multiple of second number.
Alternate method:
We can also show this solution by taking an example
Say a number is divisible by $9$, let us take that number to be $36$
We can write $36 = 9 \times 4$
Since we know $9 = 3 \times 3$
Therefore, we can write
$36 = 3 \times 3 \times 4$
Grouping together all factors other than three
$36 = 3 \times (3 \times 4) \\ 36 = 3 \times 12 \\$
Therefore, the number is divisible by $3$ as it is written in the form of multiple of three.