JEE Main 2023

Question

The number of 3-digit numbers that are divisible by either 2 or 3 but not divisible by 7 is ____. A). $514$ B). $480$ C). $506$ D). $520$

Edvaya · Accepted Answer

The correct answer is $514$. **Solution:** 3-digit numbers: 100 to 999 → 900 total. Divisible by 2: $\lfloor 999/2 \rfloor - \lfloor 99/2 \rfloor = 499 - 49 = 450$ Divisible by 3: $\lfloor 999/3 \rfloor - \lfloor 99/3 \rfloor = 333 - 33 = 300$ Divisible by 6 (both): $\lfloor 999/6 \rfloor - \lfloor 99/6 \rfloor = 166 - 16 = 150$ Divisible by 2 or 3: $450+300-150 = 600$ Divisible by 7: $\lfloor 999/7 \rfloor - \lfloor 99/7 \rfloor = 142 - 14 = 128$ Divisible by 14: $\lfloor 999/14 \rfloor - \lfloor 99/14 \rfloor = 71 - 7 = 64$ Divisible by 21: $\lfloor 999/21 \rfloor - \lfloor 99/21 \rfloor = 47 - 4 = 43$ Divisible by 42: $\lfloor 999/42 \rfloor - \lfloor 99/42 \rfloor = 23 - 2 = 21$ Divisible by (2 or 3) and 7: $64+43-21=86$ Divisible by 2 or 3, but not 7: $600-86 = 514$.

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