What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another?
1. **Representation of Numbers**:
   Each positive integer \( n \) can be uniquely written as \( n = 2^k \cdot q \) where \( k \) is a non-negative integer and \( q \) is an odd positive integer. This is essentially the prime factorization of \( n \) where \( q \) is the product of all odd prime factors of \( n \).

2. **Divisibility Condition**:
   If two numbers \( n_1 \) and \( n_2 \) have the same odd part \( q \), then one of them must be divisible by the other. This is because if \( n_1 = 2^{k_1} \cdot q \) and \( n_2 = 2^{k_2} \cdot q \) with \( k_1 \neq k_2 \), then without loss of generality, assume \( k_1 < k_2 \). Thus, \( n_1 \) divides \( n_2 \).

3. **Counting Odd Numbers**:
   We need to count the number of odd numbers between 1 and 100. The odd numbers in this range are \( 1, 3, 5, \ldots, 99 \). This is an arithmetic sequence with the first term \( a = 1 \) and common difference \( d = 2 \).

   The number of terms in this sequence can be found using the formula for the \( n \)-th term of an arithmetic sequence:
   \[
   a_n = a + (n-1)d
   \]
   Setting \( a_n = 99 \):
   \[
   99 = 1 + (n-1) \cdot 2 \implies 99 = 1 + 2n - 2 \implies 99 = 2n - 1 \implies 100 = 2n \implies n = 50
   \]
   Therefore, there are 50 odd numbers between 1 and 100.

4. **Constructing the Set**:
   Since there are 50 odd numbers, we can construct a set where each number has a unique odd part \( q \). For example, the set \( \{51, 52, 53, \ldots, 100\} \) contains 50 elements, and none of these numbers is divisible by another because they all have different odd parts.

5. **Conclusion**:
   The maximum number of elements in a set of positive integers between 1 and 100, such that no element is divisible by another, is 50.

\(\blacksquare\)

The final answer is \( \boxed{ 50 } \).