What are the values of $M$ and $N$ respectively if $M39048458N$ is divisible by both 8 and 11, where $M$ and $N$ are single-digit integers?

### Concept & Formula To determine if a large number is divisible by composite combinations, use the co-prime factors divisibility rules. For a number to be divisible by 88 (since 8 and 11 are co-prime), it must satisfy both rules: - Divisibility by 8: The number formed by the last 3 digits must be divisible by 8. - Divisibility by 11: The difference between the sum of digits at even places and odd places must be 0 or a multiple of 11. ### Step-by-Step Solution - **Given:** The number $M39048458N$ is divisible by both 8 and 11. - **Calculation (Divisibility by 8):** The last three digits form the number $58N$. For $58N$ to be divisible by 8, $N$ must be 4 because 584 is exactly divisible by 8 ($584 / 8 = 73$). - **Calculation (Divisibility by 11):** Sum of digits at even places: $8 + 4 + 4 + 9 + M = 25 + M$ Sum of digits at odd places: $4 + 5 + 8 + 0 + 3 = 20$ Difference: $(25 + M) - 20 = M + 5$ For this difference to be divisible by 11 (and since $M$ is a single-digit integer), $M + 5$ must equal 11. $$ M + 5 = 11 \implies M = 6 $$ ### Exam Strategy & Shortcut Always check the rule that affects the fewest digits first! The divisibility rule for 8 only looks at the last 3 digits ($58N$). This immediately gives $N = 4$. If you glance at the options, any choice where the second value is not 4 can be eliminated instantly, saving you from calculating the 11 divisibility rule altogether if only one option remains. ### Common Pitfall Students frequently confuse "even/odd digits" with "even/odd places". The divisibility rule for 11 requires summing the digits based on their position index (1st, 2nd, 3rd, etc.), not whether the numeric value of the digit itself is even or odd. ### Final Answer Therefore, the correct answer is 6, 4.