Find the least 6-digit number which is exactly divisible by 349.

### Concept & Logic Similar to finding a 4-digit number, to find the least $6$-digit number divisible by a divisor $D$, we start with the smallest theoretical $6$-digit base number, which is $100000$. We divide this base by the divisor to find the remainder. Because subtracting the remainder would drop us down to a $5$-digit number, we must *add* the difference between the divisor and the remainder to our base. $$ \text{Required} = 100000 + (\text{Divisor} - \text{Remainder}) $$ ### Step-by-Step Solution * **Step 1:** Set up the long division of the smallest 6-digit number by 349. $100000 \div 349$ * **Step 2:** Execute the division. $1000 \div 349 \approx 2$. ($349 \times 2 = 698$) Remainder $= 1000 - 698 = 302$ Bring down a $0$ to make $3020$. $3020 \div 349 \approx 8$. ($349 \times 8 = 2792$) Remainder $= 3020 - 2792 = 228$ Bring down the final $0$ to make $2280$. $2280 \div 349 \approx 6$. ($349 \times 6 = 2094$) Remainder $= 2280 - 2094 = 186$ * **Step 3:** Calculate the amount required to reach the next perfect multiple. Number to add $= \text{Divisor} - \text{Remainder}$ Number to add $= 349 - 186 = 163$ * **Step 4:** Add this to the base number. $100000 + 163 = 100163$ ### Exam Strategy & Shortcut **Unit Digit Check:** Look closely at the options. We are looking for a multiple of $349$ that is just above $100000$. Our quotient was roughly $286$ (giving a remainder). The next multiple must be $349 \times 287$. Look at the unit digits: $9 \times 7 = 63$. Therefore, the correct answer MUST end in a $3$. Options (a), (b), and (c) all end in $63$. Let's look at the magnitude. $349 \times 1000 \approx 349000$. $160000$ (Option c) is way too large. $101000$ (Option b) is also likely too large since $349$ is a relatively small jump from $100000$. $100163$ (Option a) represents a jump of only $163$, which is less than the divisor $349$, perfectly matching the logic of finding the *least* number. ### Common Pitfall Long division with 3-digit divisors is prone to arithmetic errors. A simple mistake in subtraction (e.g., $3020 - 2792$) cascades and guarantees an incorrect remainder, subsequently ruining the final addition. Take an extra 10 seconds to verify your subtraction steps during the long division. ### Final Answer **Therefore, the correct answer is 100163.**