A number $476**0$ is divisible by both 3 and 11. The non-zero digits in the hundred's and ten's place respectively are

## Concept & Logic To solve for missing digits using multiple divisibility rules, we must evaluate the conditions for both numbers. For a number to be divisible by **11**: The difference between the sum of its digits in odd positions and the sum of its digits in even positions must be `0` or a multiple of `11`. For a number to be divisible by **3**: The sum of all its digits must be divisible by `3`. ## Step-by-Step Solution **Given:** The number is $476ab0$, where $a$ is the digit in the hundred's place and $b$ is the digit in the ten's place. **Calculation / Deduction:** * **Step 1: Apply the divisibility rule for 11.** * Sum of digits at odd places (from right): $0 + a + 7 = a + 7$ * Sum of digits at even places (from right): $b + 6 + 4 = b + 10$ * Difference: $(a + 7) - (b + 10) = a - b - 3$ * For the number to be divisible by 11, this difference must be $0$ (since $a$ and $b$ are single digits, multiples of 11 are out of range). * Therefore, $a - b - 3 = 0 \implies a - b = 3$. * **Step 2: Apply the divisibility rule for 3.** * Sum of all digits: $4 + 7 + 6 + a + b + 0 = 17 + a + b$ * This sum $(17 + a + b)$ must be divisible by $3$. * **Step 3: Test the given options for $a$ (hundred's) and $b$ (ten's).** * *(a) 7, 4*: Check difference $7 - 4 = 3$ (Valid). Check sum $17 + 7 + 4 = 28$ (Not divisible by 3). * *(b) 5, 3*: Check difference $5 - 3 = 2$ (Invalid, must be 3). * *(c) 5, 2*: Check difference $5 - 2 = 3$ (Valid). Check sum $17 + 5 + 2 = 24$ (Divisible by 3). ## Exam Strategy & Shortcut Instead of framing equations, go straight to **Option Elimination**. Start with the rule of 11 because it creates a strict difference constraint ($a - b = 3$). Looking at the options, only (a) $7-4=3$ and (c) $5-2=3$ satisfy this. Then, mentally add those digits to the remaining sum ($17$) to see which forms a multiple of 3. $17+11=28$ (Reject), $17+7=24$ (Accept!). ## Common Pitfall Students often misread "hundred's and ten's place respectively" and assign the values backwards, or they sum the alternating digits starting from the wrong side. While starting from the left vs. right won't change the absolute difference, making a simple arithmetic error in $(a+7) - (b+10)$ is the most common trap. Always write the steps out carefully. ## Final Answer **Therefore, the correct answer is 5, 2.**