The smallest 6-digit number exactly divisible by 111 is

### Concept & Formula To find the smallest $n$-digit number perfectly divisible by a divisor $D$, start with the smallest possible $n$-digit base number. Find the remainder upon division, and add the difference between the divisor and the remainder to bridge the gap to the next multiple. $$ \text{Required Number} = \text{Base} + (\text{Divisor} - \text{Remainder}) $$ ### Step-by-Step Solution * **Step 1:** Identify the absolute smallest 6-digit number. The base number is $100000$. * **Step 2:** Execute long division to find the remainder. $100000 \div 111$ We know $111 \times 9 = 999$. So, $111 \times 900 = 99900$. Subtracting this from the base: $100000 - 99900 = 100$. The remainder is $100$. * **Step 3:** Calculate the amount required to reach the next perfect multiple. Number to add $= \text{Divisor} - \text{Remainder}$ Number to add $= 111 - 100 = 11$. * **Step 4:** Add this value to the original base number. $100000 + 11 = 100011$. ### Exam Strategy & Shortcut **Option Elimination:** Look at the options and apply the divisibility rule for 3 (since 111 is divisible by 3, the answer must also be). (a) $111111 \rightarrow$ Sum is 6 (Divisible, but is it the smallest?) (b) $110011 \rightarrow$ Sum is 4 (Not divisible) (c) $100011 \rightarrow$ Sum is 3 (Divisible) (d) $110101 \rightarrow$ Sum is 4 (Not divisible) Comparing the valid candidates, $100011$ is significantly smaller than $111111$. A quick mathematical check ($100011 = 99900 + 111$) confirms it is built from exact multiples. ### Common Pitfall Students often incorrectly assume a number like $111111$ is the smallest simply because it repeats the digits of the divisor and feels conceptually "clean". Never assume visual patterns bypass mathematical rules; always start your logic from the fundamental numerical base $100000$. ### Final Answer **Therefore, the correct answer is 100011.**