To solve this problem, let's analyze the set $A = \{91, 92, 93, \ldots, 180\}$ and the properties of subset $B$.

1. Step 1: Find the Range of Numbers and Remainders Modulo 9

   The elements in $A$ start from 91 and go up to 180. These numbers form an arithmetic sequence with a common difference of 1. The length of $A$ is: $180 - 91 + 1 = 90$ So, $A$ contains 90 elements.

   Now, let's find the remainder when each element in $A$ is divided by 9. We will analyze each possible remainder (from 0 to 8) because two elements in $B$ cannot have a sum that is a multiple of 9.

2. Step 2: Determine Feasible Elements Based on Modulo 9 Compatibility

   Since we cannot have any two elements in $B$ whose sum is divisible by 9, we must choose elements such that if an element has remainder $r$, we avoid elements with remainder $9 - r$.

   Here's how we can pair the remainders:

   • $r = 0$ (cannot pair with itself in $B$)
   • $r = 1$ pairs with $r = 8$
   • $r = 2$ pairs with $r = 7$
   • $r = 3$ pairs with $r = 6$
   • $r = 4$ pairs with $r = 5$
   • $r = 8$ pairs with $r = 1$, etc.

   To maximize the size of $B$, we choose one element from each of these pairs.

3. Step 3: Count the Frequency of Each Remainder in Set $A$

   Calculating each remainder's occurrence from 91 to 180, we find that each remainder (from 0 to 8) occurs exactly 10 times.

4. Step 4: Select Maximum Elements Based on Non-pairing Remainders

   To avoid sums divisible by 9, we can select:

   • One element from each pair of remainders, choosing a maximum of 5 elements per pair.
   • This results in $5 \times 8 = 40$.

5. Conclusion: Maximum Number of Elements in $B$

   The maximum number of elements in $B$ that satisfy the condition is therefore 40.

The correct answer is:

• None of the given choices directly match. However, if you can verify further or see a similar pattern, confirm the possible maximum number as 40 elements.

Would you like more clarification on any part of this solution?

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Here are some related questions you might consider:

1. How would the solution change if the range of $A$ were different?
2. How do we calculate the exact distribution of remainders in any sequence?
3. What if the condition was for sums to be non-divisible by another number, like 7 or 8?
4. Can this method apply to subsets with other operations, like products?
5. How do subsets with restricted sum properties apply in probability problems?

Tip: When dealing with modular arithmetic in sets, organizing elements by remainder often simplifies subset problems with divisibility restrictions.