The smallest 4-digit number exactly divisible by 7 is

### Concept & Strategy To find the smallest $n$-digit number exactly divisible by a given divisor $d$, we start with the absolute smallest $n$-digit number base. We then find the remainder when this base is divided by $d$. The number we need to *add* to the base to reach the next perfect multiple is $(d - \text{remainder})$. Formula: $$ \text{Required Number} = \text{Base} + (\text{Divisor} - \text{Remainder}) $$ ### Step-by-Step Solution * **Step 1:** Identify the smallest 4-digit number. The smallest 4-digit number is $1000$. * **Step 2:** Divide this base number by the given divisor (7) to find the remainder. $1000 \div 7$ $7 \times 142 = 994$ Remainder $= 1000 - 994 = 6$ * **Step 3:** Calculate how much needs to be added to make it perfectly divisible. Number to add $= \text{Divisor} - \text{Remainder}$ Number to add $= 7 - 6 = 1$ * **Step 4:** Add this value to the original base number. $1000 + 1 = 1001$ ### Exam Strategy & Shortcut **Option Elimination via Direct Division:** Since the options are close to $1000$, start dividing the smallest options directly by $7$. Option (a) is $1001$. Let's apply the divisibility rule for 7, or simply divide: $100 \div 7 = 14$ remainder $2$. Bring down $1$ to make $21$. $21 \div 7 = 3$. It divides perfectly. Since it's the smallest option, it must be the answer. No complex formulas required! ### Common Pitfall A very common mistake is simply subtracting the remainder ($6$) from the base number ($1000$). While $1000 - 6 = 994$ is indeed divisible by 7, it is a 3-digit number. The question specifically asks for a 4-digit number, meaning we must *add* to reach the next multiple. ### Final Answer **Therefore, the correct answer is 1001.**