If the number $357*25*$ is divisible by both 3 and 5, then the missing digits in the unit's place and the thousandth's place respectively are

## Concept & Logic To find multiple missing digits, tackle the rule that restricts the end of the number first (like 2, 5, or 10), then apply summation rules (like 3 or 9). * **Divisibility by 5:** The unit digit must be strictly `0` or `5`. * **Divisibility by 3:** The sum of all digits must be a multiple of `3`. ## Step-by-Step Solution **Given:** The number is $357x25y$, where $x$ is the digit in the thousand's place (noted as thousandth's in the question) and $y$ is the digit in the unit's place. The question asks for the values of $(y, x)$ respectively. **Calculation / Deduction:** * **Step 1: Check the unit's place ($y$) for divisibility by 5.** For the number to be divisible by 5, the unit digit $y$ must be either $0$ or $5$. * **Step 2: Formulate the rule for divisibility by 3.** Sum of the digits: $3 + 5 + 7 + x + 2 + 5 + y = 22 + x + y$. This sum $(22 + x + y)$ must be a multiple of 3. * **Step 3: Evaluate the given options for $(y, x)$ respectively.** * *(a) 0, 6*: $y=0$, $x=6$. Condition for 5 is met. Sum $= 22 + 6 + 0 = 28$. $28$ is not divisible by 3. (Reject) * *(b) 5, 1*: $y=5$, $x=1$. Condition for 5 is met. Sum $= 22 + 1 + 5 = 28$. $28$ is not divisible by 3. (Reject) * *(c) 5, 4*: $y=5$, $x=4$. Condition for 5 is met. Sum $= 22 + 4 + 5 = 31$. $31$ is not divisible by 3. (Reject) Since none of the provided pairs yield a digit sum that is a multiple of 3, none of the options are correct. ## Exam Strategy & Shortcut Always scan the required order of the options. The prompt asks for **unit's place** and then **thousand's place**, meaning the options are ordered $(y, x)$. Check the first digit of every option immediately to see if it satisfies the rule of 5 (it must be 0 or 5). Since all options start with 0 or 5, you quickly transition to adding the two digits to the base sum of $22$. $22 + 0 + 6 = 28$ (No) $22 + 5 + 1 = 28$ (No) $22 + 5 + 4 = 31$ (No) You can solve this in under 15 seconds without writing anything down. ## Common Pitfall The most common mistake is ignoring the word "respectively" and swapping the variables, assuming the options are in left-to-right order $(x, y)$. However, even if a student swapped them here, they would calculate the exact same sums ($x+y = y+x$) and still correctly arrive at "None of these". The real pitfall is math anxiety causing a student to second-guess themselves when none of the primary options work. Trust your math! ## Final Answer **Therefore, the correct answer is None of these.**