JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Permutations and Combinations

Among the $900$ three-digit numbers, those divisible by $2$ number $450$ and those by $3$ number $300$, with $150$ divisible by $6$, so $|A\cup B|=450+300-150=600$. Of these, the ones also divisible by $7$ are the multiples of $14$ (there are $64$) and of $21$ (there are $43$), less the multiples of $42$ (there are $21$), giving $64+43-21=86$. Removing them leaves $600-86=514$ such numbers.