If $A$ has $m$ members and $B$ has $n$ members, then $2^m - 2^n = 2016 = 2^5 \cdot 63$. Find the number of members in $A \cup B$.
Step-by-Step Solution
Key Concept: Factorize the exponential equation $2^m - 2^n$ to separate powers of 2 and solve for $m$ and $n$ individually using prime factorization.
We have $2^m - 2^n = 2016 = 2^5 \cdot 63$. Factoring the left side: $2^n(2^{m-n} - 1) = 2^5 \cdot 63$. This means $n = 5$ and $2^{m-n} - 1 = 63$, so $2^{m-n} = 64 = 2^6$, giving $m - n = 6$. Therefore $m = 11$. The number of members in $A \cup B = m + n - |A \cap B| = 11 + 5 - 3 = 13$, where we use the inclusion-exclusion principle and the constraint that the overlap has 3 elements.
Correct Answer: 13