$136 \times 12 \times 8 = x$

### Concept & Strategy This is a sequential multiplication problem. Because multiplication is associative, you can group the numbers in whatever order makes the calculation easiest. The goal is to reach a number close to a base of $100$ to utilize mental math. $$ A \times B \times C = A \times (B \times C) $$ ### Step-by-Step Solution * **Given:** The expression $136 \times 12 \times 8$. * **Step 1:** Group the smaller numbers first to create a single multiplier. $12 \times 8 = 96$ * **Step 2:** The expression simplifies to $136 \times 96$. * **Step 3:** Use the distributive property, recognizing that $96$ is close to $100$. $136 \times (100 - 4)$ * **Step 4:** Multiply across the bracket. $(136 \times 100) - (136 \times 4)$ $13600 - 544$ * **Step 5:** Perform the subtraction. $13600 - 544 = 13056$ * **Step 6:** Compare the correct result ($13056$) with the given options: (a) 12066, (b) 13046, (c) 13064, (d) 13066. Since it does not appear, the correct choice is "None of these". ### Exam Strategy & Shortcut Use **Unit Digit Analysis** to eliminate options rapidly. Multiply the unit digits of the given numbers: $6 \times 2 \times 8$. $6 \times 2 = 12$ (ends in $2$). $2 \times 8 = 16$ (ends in $6$). The final answer MUST end in $6$. This instantly eliminates option (c) $13064$. Next, estimate the magnitude: $136 \times 100 = 13600$. Since we are multiplying by $96$ (which is $4$ less than $100$), the answer must be slightly less than $13600$. $136 \times 4$ is roughly $500$. So, $13600 - 500 = 13100$. None of the remaining options (12066, 13046, 13066) match the exact calculation of $13056$. ### Common Pitfall When students calculate $13056$ and see options like $13046$ or $13066$, they often assume they made a small arithmetic error and guess one of the close options. Trust your precise math and boldly select "None of these" if your calculation is solid. ### Final Answer Therefore, the correct answer is **None of these**.