JEE Main 2023 — Mathematics Algebra

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

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

$$A={100,102,...,998}$$A=100,102,...,998

$$\Rightarrow n(A)=450$$⇒n(A)=450

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

$$n(B)=300$$n(B)=300

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

${102,108,...996}$102,108,...996 i.e, $150$150

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

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

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

Required number

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

Hence, $514$514 three-digit number are there which are divisible by $2\text{or}3\text{but not divisible by}7$2or3but not divisible by7.