Mathematics - Sequence & Series Question with Solution | TestHub

We know that total number of three-digit number will be $900$,

Now let $n\left(A\right)$ be number of three-digit number which are divisible by $2$, i.e,

$A=\left\{100,102,...,998\right\}$

$\Rightarrow n\left(A\right)=450$

$n\left(B\right)$ be number of three-digit number which are divisible by $3$, i.e, $B=\left\{102,105,...,999\right\}$

$n\left(B\right)=300$

Numbers divisible by both $2$ and $3$ is

$\left\{102,108,...996\right\}$ i.e, $150$

Numbers divisible by both $2$ and $7$ are $\left\{112,126,...,994\right\}$ i.e., $64$ numbers.

Numbers divisible by both $3$ and $7$ is $\left\{105,126,...987\right\}$ i.e, $43$ numbers.

Numbers divisible by $2,3 \& 7$ is $\left\{126,168,...,966\right\}$ i.e, $21$ numbers.

Required number

$=450+150-64-43+21=514$

Hence, $514$ three-digit number are there which are divisible by $2 \text{or} 3 \text{but not divisible by} 7$.