Let $ M \ge 3$ be an integer and let $ S \equal{} \{3,4,5,\ldots,m\}$. Find the smallest value of $ m$ such that for every partition of $ S$ into two subsets, at least one of the subsets contains integers $ a$, $ b$, and $ c$ (not necessarily distinct) such that $ ab \equal{} c$.

[b]Note[/b]: a partition of $ S$ is a pair of sets $ A$, $ B$ such that $ A \cap B \equal{} \emptyset$, $ A \cup B \equal{} S$.
To solve this problem, we need to find the smallest value of \( m \) such that for every partition of \( S = \{3, 4, 5, \ldots, m\} \) into two subsets, at least one of the subsets contains integers \( a \), \( b \), and \( c \) (not necessarily distinct) such that \( ab = c \).

1. **Initial Consideration**:
   - We start by considering the set \( S \) and the condition that for any partition of \( S \) into two subsets \( A \) and \( B \), one of these subsets must contain \( a \), \( b \), and \( c \) such that \( ab = c \).

2. **Bounding for \( m \leq 242 \)**:
   - If \( m \leq 242 \), we can construct a partition that avoids the condition \( ab = c \). Consider the partition:
     \[
     A = \{3, 4, \ldots, 8, 81, 82, \ldots, m\}
     \]
     \[
     B = \{9, 10, \ldots, 80\}
     \]
   - In this partition, \( A \) contains numbers that are either too small or too large to form a product that lies within \( S \), and \( B \) contains numbers that are not products of any two numbers in \( A \).

3. **Constructing a Bad Set for \( m = 243 \)**:
   - We now attempt to construct a partition for \( m = 243 \) and show that it is impossible to avoid the condition \( ab = c \).
   - Without loss of generality (WLOG), let \( 3 \in A \). Since \( 3^2 = 9 \), we must have \( 9 \in B \).
   - Since \( 9^2 = 81 \), we must have \( 81 \in A \).
   - Because \( 3 \cdot 27 = 81 \), we must have \( 27 \in B \).
   - However, \( 3 \cdot 81 = 243 \) and \( 9 \cdot 27 = 243 \), implying that \( 243 \) cannot belong to either set \( A \) or \( B \) without violating the condition \( ab = c \).

4. **Conclusion**:
   - Since it is impossible to partition \( S \) for \( m = 243 \) without one of the subsets containing \( a \), \( b \), and \( c \) such that \( ab = c \), we conclude that the smallest value of \( m \) is \( \boxed{243} \).