Question

Find the sum of all two digit positive integers which are divisible by 3 but not by 2.

Answer 1

Two digit numbers which are divisible by $3$ but not by $2$-

$15,21,27,.....,99$

First term $\left(a\right)=15$

Common difference $\left(d\right)=6$

Last term $\left(l\right)=99$

No. of terms $\left(n\right)=?$

As we know that $n^{th}$ term in an A.P. is given as-

$a_n=a+(n-1)d$

$99=15+(n-1)6$

$n-1=\frac{99-15}{6}$

$n=14+1=15$

Again sum of $n$ terms in an A.P. is given as-

$S_n=\frac{n}{2}(a+l)$

$\therefore S_n=\frac{15}{2}(15+99)$

$\Rightarrow S_n=\frac{15}{2}\times 114=855$

Hence the sum of all two digit numbers which are divisible by $3$ but not by $2$ is $855$.