$6 \times 66 \times 666 = x$

### Concept & Logic This is a sequential multiplication problem involving repeating digits. The most efficient way to handle it is to multiply the smaller factors first, and then apply distribution techniques to the final large multiplication to reduce cognitive load. ### Step-by-Step Solution * **Given:** The expression $6 \times 66 \times 666$. * **Step 1:** Multiply the two smaller numbers. $6 \times 66 = 396$ * **Step 2:** Substitute back into the expression. $396 \times 666$ * **Step 3:** Use the distributive property to make mental math easier. Notice that $396$ is very close to $400$. $(400 - 4) \times 666$ * **Step 4:** Distribute the multiplication across the terms. $(400 \times 666) - (4 \times 666)$ * **Step 5:** Calculate the individual parts. $400 \times 666 = 266400$ $4 \times 666 = 2664$ * **Step 6:** Perform the final subtraction. $266400 - 2664 = 263736$ ### Exam Strategy & Shortcut You can eliminate options rapidly using the **Unit Digit** and **Last Two Digits** methods. 1. **Unit digit:** $6 \times 6 \times 6 = 216$. The answer must end in $6$. This instantly eliminates option (b) $263763$. 2. **Last two digits:** From $396 \times 666$, we can use $(400 - 4) \times 666$. The last two digits will be determined entirely by $-4 \times 66 = -264$. Subtracting this from a hundreds base (like $400 - 264$), the last two digits will be $100 - 64 = 36$. Options (a), (c), and (d) all end in $36$. 3. **Digital Sum:** The digital root of $6$ is $6$. The digital root of $66$ is $3$. The digital root of $666$ is $9$. Any number multiplied by a multiple of $9$ results in a digital sum of $9$. (a) $2+6+3+3+7+6 = 27 = 9$. (c) $2+6+3+7+3+6 = 27 = 9$. At this point, the $400 - 4$ distribution shortcut shown in the main steps is the definitive way to confirm $263736$ without full column multiplication. ### Common Pitfall Writing this out vertically as $666 \times 66$ and then multiplying by $6$ again is highly prone to alignment and carry-over mistakes due to the repeating '6's confusing the eye. Leveraging the $(400 - 4)$ base is much safer. ### Final Answer Therefore, the correct answer is **263736**.