Question

Let \( P = \{2, 3, 4, \ldots, 100\} \) and \( Q = \{101, 102, 103, \ldots, 200\} \). How many elements of \( Q \) are there such that they do not have any element of \( P \) as a factor?

• A. 20
• B. 24
• C. 21
• D. 23

Hint:

When no factors are allowed from a set of integers, think in terms of co-prime numbers — often prime numbers.

Answer 1

The Correct Option is C

Any number from 101 to 200 that is not divisible by any element from 2 to 100 must be a prime number.
So, we want to count how many prime numbers lie in the interval \([101, 200]\).
List of such primes: \[ 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 \] Count = \boxed{21}