What number should replace $M$ in this multiplication problem? $$ \begin{matrix} & & 3 & M & 4 \\ & \times & & & 4 \\ \hline & 1 & 2 & 1 & 6 \end{matrix} $$

### Concept & Logic This problem involves missing digits in multiplication. We need to find a single digit $M$ that satisfies the algorithm. We can solve this either by analyzing place-value carry-overs or by using the inverse operation (division). ### Step-by-Step Solution * **Given:** $3M4 \times 4 = 1216$. * **Units Place:** $4 \times 4 = 16$. We write down $6$ and carry over $1$ to the tens place. * **Tens Place:** We multiply the unknown $M$ by $4$, add the carry, and the result must end in $1$. $$4 \times M + 1 \text{ (carry)} = \text{number ending in } 1$$ This means $4 \times M$ must end in $0$. The only single digits that yield a product ending in $0$ when multiplied by $4$ are $0$ and $5$. * **Hundreds Verification:** If $M = 0$: $4 \times 0 + 1 = 1$. The carry to the hundreds place is $0$. Then $4 \times 3 = 12$. The final number is $1216$. This matches perfectly. If $M = 5$: $4 \times 5 + 1 = 21$. The carry to the hundreds place is $2$. Then $4 \times 3 + 2 = 14$. The final number would be $1416$, which is incorrect. ### Exam Strategy & Shortcut **Inverse Operation:** You have the final product and one of the factors. Do not waste time with place-value equations; simply divide the product by the known factor to find the missing number! $$1216 \div 4 = 304$$ By directly comparing $304$ with $3M4$, it is immediately obvious that $M = 0$. ### Common Pitfall Guessing and checking every single digit from $1$ to $9$ in the multiplication format. While it works, it burns valuable exam time. Recognizing that division is the fastest path is key. ### Final Answer Therefore, the correct answer is **0**.