If $37X3$ is a four-digit natural number divisible by 7, then the place marked as $X$ must have the value

## Concept & Logic While there is a formal divisibility rule for 7 (double the last digit and subtract it from the remaining number), for a small four-digit number with one unknown, it is often faster to use **Place Value Expansion** and standard remainder algebra. ## Step-by-Step Solution **Given:** The number $37X3$ is completely divisible by 7. **Calculation / Deduction:** * **Step 1: Expand the number using place values.** $37X3$ can be written mathematically as $3700 + 10X + 3$, which simplifies to $3703 + 10X$. * **Step 2: Find the remainder of the known portion when divided by 7.** Let's divide $3703$ by $7$. $3500 \div 7 = 500$ Remaining: $3703 - 3500 = 203$. $203 \div 7 = 29$ exactly ($7 \times 20 = 140$, $7 \times 9 = 63$, $140+63=203$). Since $3703$ is perfectly divisible by $7$ (it equals $7 \times 529$), its remainder is $0$. * **Step 3: Solve for $X$.** Since $3703 + 10X$ must be divisible by 7, and $3703$ is already divisible by 7, the remaining part, $10X$, must also be a multiple of 7. For $10 \times X$ to be a multiple of 7, $X$ itself must be a multiple of 7 (since 10 and 7 are co-prime). Because $X$ is a single digit (0-9), the only valid multiples of 7 are $0$ and $7$. * **Step 4: Check the options.** Looking at the choices (0, 3, 5, 9), only $0$ is available. (If 7 were an option, both 3703 and 3773 would be valid answers, but only 0 is provided). ## Exam Strategy & Shortcut Instead of expanding algebraically, you can test the options rapidly using the **Osculator Method** (Rule of 7: drop unit digit, multiply by 2, subtract). Let's test Option (a) $X=0$: Number is $3703$. Drop 3, double it (6). Subtract from 370: $370 - 6 = 364$. Drop 4, double it (8). Subtract from 36: $36 - 8 = 28$. $28$ is a known multiple of 7. The first option works, allowing you to move on immediately! ## Common Pitfall A common trap is assuming $X$ cannot be $0$ because it "hides" the tens place, making students lean towards non-zero options. Remember that $0$ is a perfectly valid digit in the middle of a natural number, and $0 \times 10 = 0$ is a perfectly valid multiple of any number (including 7). ## Final Answer **Therefore, the correct answer is 0.**