What least value must be assigned to $*$ so that the number $197*5462$ is divisible by $9$?

### Concept & Logic This problem relies on the standard divisibility rule for $9$. A number is divisible by $9$ if the total sum of its digits is a multiple of $9$. ### Step-by-Step Solution **Given:** The number is $197*5462$. Let the unknown digit $*$ be $x$. **Calculation:** Step 1: Calculate the sum of the known digits. Sum $= 1 + 9 + 7 + x + 5 + 4 + 6 + 2$ Sum $= 34 + x$ Step 2: Determine the least value of $x$. For $(34 + x)$ to be divisible by $9$, it must equal the next multiple of $9$ that is greater than or equal to $34$. The multiples of $9$ are $9, 18, 27, 36, 45...$ The immediate next multiple is $36$. So, $34 + x = 36$ $x = 36 - 34 = 2$ ### Exam Strategy & Shortcut Use the ''casting out nines'' technique to save time and reduce mental fatigue. Ignore any $9$s or combinations of digits that sum to $9$. For $197*5462$: - Ignore $9$ - Ignore $7 + 2 = 9$ - Ignore $5 + 4 = 9$ You are left with $1 + 6 + x = 7 + x$. For $7 + x$ to be divisible by $9$, $x$ must be $2$. ### Common Pitfall Students often make simple arithmetic errors when adding a long string of numbers sequentially (like getting $33$ instead of $34$). Using the cancellation method completely prevents this pitfall by keeping the numbers small. ### Final Answer Therefore, the correct answer is **2**.