Question
How many two-digit numbers are divisible by $3 ?$
✓
Answer
Numbers divisible by $3$ are multiples of $3 .$
$3,6,9,12,15 \ldots$
The smallest two-digit number divisible by $3$ is $12$. And the largest $2-$digit number divisible by $3$ is $99 .$
So, the series of the $2-$digit multiples of $3$ starts with $12$ and ends with $99$ . The difference between the numbers is $3$ .
Therefore, the $A.P.$ is $12,15,18,21,24 \ldots 90,93,96,99$.
In the sequence, the first term, $a=12$.
The last term, $l=99$.
The common difference, $d=3$. The $n^{\text {th }}$ term is $a_n=99$.
Now, $a_n=a_1+(n-1) d$
$\Rightarrow 99=12+(n-1) 3 \Rightarrow 99=12+3 n-3$
$\Rightarrow 99=3 n+9 \Rightarrow 3 n=90$
$\Rightarrow n=30$
Therefore, there are $30$ two$-$digit numbers divisible by $3 .$
$3,6,9,12,15 \ldots$
The smallest two-digit number divisible by $3$ is $12$. And the largest $2-$digit number divisible by $3$ is $99 .$
So, the series of the $2-$digit multiples of $3$ starts with $12$ and ends with $99$ . The difference between the numbers is $3$ .
Therefore, the $A.P.$ is $12,15,18,21,24 \ldots 90,93,96,99$.
In the sequence, the first term, $a=12$.
The last term, $l=99$.
The common difference, $d=3$. The $n^{\text {th }}$ term is $a_n=99$.
Now, $a_n=a_1+(n-1) d$
$\Rightarrow 99=12+(n-1) 3 \Rightarrow 99=12+3 n-3$
$\Rightarrow 99=3 n+9 \Rightarrow 3 n=90$
$\Rightarrow n=30$
Therefore, there are $30$ two$-$digit numbers divisible by $3 .$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The following distribution shows the transport expenditure of $100$ employees:
Find the mode of the distribution
→| Expenditure $($in Rs.$):$ | $200-400$ | $400-600$ | $600-800$ | $800-1000$ | $1000-1200$ |
| Number of employees: | $21$ | $25$ | $19$ | $23$ | $12$ |
Evaluate the following:
Verify the following:
$\sin60^\circ\cos30^\circ-\cos60^\circ\sin30^\circ=\sin30^\circ$
→Verify the following:
$\sin60^\circ\cos30^\circ-\cos60^\circ\sin30^\circ=\sin30^\circ$
Find the sum of the following APs:
$\frac{1}{15},\frac{1}{12},\frac{1}{10},\ ....$ to 11 terms.
→$\frac{1}{15},\frac{1}{12},\frac{1}{10},\ ....$ to 11 terms.
The graph of y = p(x) in a figure given below, for some polynomial p(x). Find the number of zeroes of p(x).
Find a quadratic polynomial, the sum and product of whose zeroes are $\sqrt { 2 } , \frac { 1 } { 3 }$ respectively.
→Prove that:
$\frac{\cos80^\circ}{\sin10^\circ}+\cos59^\circ\text{cosec }31^\circ=2$
→$\frac{\cos80^\circ}{\sin10^\circ}+\cos59^\circ\text{cosec }31^\circ=2$
What is the value of $(1+\cot^2\theta)\sin^2\theta?$
→It is given that E and F are points on the sides PQ and PR respectively of a $\triangle$PQR. For PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm, state whether EF||QR.
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
Prove that: $\frac{1-\cos A}{1+\cos A}=(\cot A-\operatorname{cosec} A)^2$
→