The least number, which must be added to the greatest 6-digit number so that the sum may be exactly divisible by 327 is

### Concept & Logic When a question asks for the least number to be *added* to a base number to make it exactly divisible, we must determine how far the current remainder is from reaching the next full multiple of the divisor. We must bridge the gap between the remainder and the divisor itself. $$ \text{Number to add} = \text{Divisor} - \text{Remainder} $$ ### Step-by-Step Solution * **Step 1:** Identify the greatest 6-digit number, which is $999999$. * **Step 2:** Set up and execute long division by $327$. $999 \div 327 = 3$ ($327 \times 3 = 981$). Remainder is $18$. Bring down $9$ to make $189$. Since $189 < 327$, the quotient is $0$. Bring down the next $9$ to make $1899$. $1899 \div 327 = 5$ ($327 \times 5 = 1635$). Remainder is $264$. Bring down the final $9$ to make $2649$. $2649 \div 327 = 8$ ($327 \times 8 = 2616$). Remainder is $33$. * **Step 3:** Note the divisor ($327$) and the final remainder ($33$). * **Step 4:** Calculate the required addition. Number to add $= 327 - 33 = 294$. ### Exam Strategy & Shortcut **Divisibility Rule of 3:** The divisor $327$ is a multiple of $3$ (since $3+2+7 = 12$). The base number $999999$ is obviously a multiple of $3$. If we add an unknown number $x$ to $999999$, the sum ($999999 + x$) must be divisible by $327$, and therefore must be divisible by $3$. Since $999999$ is already a multiple of $3$, the number $x$ we add MUST also be a multiple of $3$. Check the options: * (a) $194 \rightarrow 1+9+4 = 14$ (No) * (b) $264 \rightarrow 2+6+4 = 12$ (Yes) * (c) $292 \rightarrow 2+9+2 = 13$ (No) * (d) $294 \rightarrow 2+9+4 = 15$ (Yes) This instantly narrows the field down to two choices, drastically reducing the time spent if you need to guess or do partial division. ### Common Pitfall The most common mistake is stopping at the remainder ($33$) and selecting it as the answer, or selecting a distractor option that represents a partial calculation step (like $264$, which appeared as an intermediate remainder during step 2). Always apply the formula $\text{Divisor} - \text{Remainder}$ for "must be added" questions. ### Final Answer **Therefore, the correct answer is 294.**