What least value must be given to $n$ so that the number $6135n2$ becomes divisible by 9?

### Concept & Formula The divisibility rule for $9$ states that a number is perfectly divisible by $9$ if and only if the sum of all its digits is a multiple of $9$. ### Step-by-Step Solution * **Given Number:** $6135n2$, where $n$ is a single-digit integer ($0 \le n \le 9$). * Calculate the sum of the known digits: $$\text{Sum} = 6 + 1 + 3 + 5 + n + 2$$ $$\text{Sum} = 17 + n$$ * For the entire number to be divisible by $9$, the expression $(17 + n)$ must equal a multiple of $9$. * The multiples of $9$ are $9, 18, 27, 36, \dots$ * Since $n$ must be a positive, single-digit value, the closest multiple of $9$ greater than or equal to $17$ is $18$. * Set up the equation to find the least value: $$17 + n = 18$$ $$n = 18 - 17 = 1$$ ### Exam Strategy & Shortcut **Casting out Nines:** Ignore any digits that sum to $9$ while adding. In $6135n2$: $(6+3)=9$ (cast out). Remaining digits: $1 + 5 + 2 = 8$. We need $8 + n$ to equal $9$. Therefore, $n = 1$. This saves precious calculation time. ### Common Pitfall A common mistake is skipping over zero when checking for least values. Always remember $0$ is a valid digit. If the sum had been exactly $18$ without $n$, the least value for $n$ would be $0$, not $9$. However, here $17+n=18$, so $n=1$. ### Final Answer Therefore, the correct answer is **1**.