If 78*3945 is divisible by 11, where * is a digit, then * is equal to

### Concept & Formula The divisibility rule for $11$ requires that the difference between the sum of the digits in odd positions and the sum of the digits in even positions must be either $0$ or a multiple of $11$ ($11, 22, 33$, etc.). ### Step-by-Step Solution * **Given:** The number is $78*3945$. Let us label the positions from left to right. * **Summing the alternating digits:** Sum of digits at odd places ($1$st, $3$rd, $5$th, $7$th): $7 + * + 9 + 5 = 21 + *$ Sum of digits at even places ($2$nd, $4$th, $6$th): $8 + 3 + 4 = 15$ * **Calculation:** According to the rule, the difference between these two sums must be $0$ or a multiple of $11$. Difference: $(21 + *) - 15 = 6 + *$ Since $*$ is a single digit from $0$ to $9$, the maximum value for $6 + *$ is $6 + 9 = 15$. Therefore, the difference cannot reach $22$. The only valid multiple of $11$ it can equal is exactly $11$. $$6 + * = 11$$ $$* = 11 - 6 = 5$$ ### Exam Strategy & Shortcut Use mental running differences instead of writing out two separate sums. Proceeding left to right: $7 - 8 + * - 3 + 9 - 4 + 5 = 6 + *$ Set the result equal to $11$ (since $6$ is greater than $0$ but less than $11$). This quickly gives $* = 5$ without writing down long equations. ### Common Pitfall Students often confuse the positions, accidentally grouping the missing variable $*$ into the wrong set. Additionally, some test-takers set the equation to $0$ ($6 + * = 0$) and get stuck, forgetting that the difference can also be $11$. ### Final Answer Therefore, the correct answer is **5**.