Question

# How many numbers of two digits are divisible by $3$ ?

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Hint: Select the lowest and highest two digit terms divisible by $3$. To see whether the series is an A.P or not. Then if it is in A.P solve it by taking the ${{n}^{th}}$ term of the A.P. Find the value of $n$. You will get the answer.

We use ten digits in the way we count. Sometimes we use one digit to represent a number and sometimes we use more. In this lesson, you'll learn how many digits are in any given numeral, and recognize a digit when you see it. You will also begin learning about place value.
This is the numeral: $51$.These are the digits in that numeral: $5$ and $1$. The numeral has two digits because two symbols, or digits, make up the numeral $51$. What about this numeral ? $452$. This numeral has three digits: $4,5$ and $2$.
That $7$ is a special kind of symbol. It's called a digit. We have ten digits we use to make up all numerals. A numeral is a number written down. These digits are $0,1,2,3,4,5,6,7,8$ and $9$. That's it! You can make any numeral you want out of those ten digits.
Just like a red light means 'stop', a $7$ means 'seven'. It's a single symbol that represents a numeral. Yes, just like this is a symbol to stop.
We know, first two digit number divisible by $3$ is $12$ and the last two digit number divisible by $3$ is $99$. Thus, we get $12,15,......,99$.
So the lowest two digit number divisible by $3$ is $12$.
Highest two digit number divisible by $3$ is $99$.
So we can see the difference between the numbers that are divisible by 3, is $3$.
So the above series is in A.P.
We have to find it in terms of $n$.
And here$a=12,d=3,{{a}_{n}}=99$
Thus, the ${{n}^{th}}$term of A.P is :
${{a}_{n}}=a+(n-1)d$
Where,
$a=$First-term
$d=$ Common difference
$n=$ number of terms
${{a}_{n}}={{n}^{th}}$term
So now applying the formula for 99, we get,
$99=12+(n-1)3$
Simplifying further we get,
\begin{align} & 99-12=(n-1)3 \\ & 87=3n-3 \\ & 90=3n \\ & n=30 \\ \end{align}
So we get $n=30$.
Therefore, the number of two digits divisible by $3$ are $30$.

Note: Read the question properly. Also, we should know the lowest and highest two digit terms divisible by $3$. So the concepts related to A.P should be clear. Here we have used the concept of A.P that is we have used ${{n}^{th}}$ term of A.P which is ${{a}_{n}}=a+(n-1)d$.