UPSC Prelims 2025 CSAT, question 25

Solution.

Answer: (c) 22

1. Since $9$ is a multiple of $3$, a number is not divisible by any of $2, 3, 5, 7,$ and $9$ if and only if it is not divisible by any of the four primes $2, 3, 5,$ and $7$.
2. Every composite number $\le 100$ has a prime factor $\le \sqrt{100} = 10$, which must be $2, 3, 5,$ or $7$; therefore, the surviving numbers are $1$ together with all prime numbers between $11$ and $100$.
3. There are $25$ primes up to $100$; excluding the $4$ primes $(2, 3, 5, 7)$ leaves $21$ primes, and adding the number $1$ gives $21 + 1 = 22$ numbers.

Trap: Remember to include the number $1$ (which is not divisible by $2, 3, 5,$ or $7$), shifting the count from $21$ to $22$.