Which of the following numbers is divisible by both 9 and 11? A) 277218 B) 10098 C) 12345 D) 181998

Understand the Problem

The question is asking to identify which number from the given options is divisible by both 9 and 11. To solve it, we need to check the divisibility of each number by these two factors.

Answer

The correct answer is the number that satisfies $ \text{Option} \div 99 = \text{Whole number} $.

Steps to Solve

Identify the LCM of 9 and 11

To check if a number is divisible by both 9 and 11, we first calculate the least common multiple (LCM) of these two numbers. Since 9 and 11 are coprime (they have no common factors other than 1), the LCM can be found by:

$$ \text{LCM}(9, 11) = 9 \times 11 = 99 $$

Check divisibility of each option by 99

Now, take each option provided in the question and divide it by 99 to check if it results in a whole number. A number is divisible by 99 if:

$$ \text{Option} \div 99 = \text{Whole number} $$

List the options and perform calculations

For each number in the options, calculate:

If the number is divisible by 99, it is our answer.
Use long division or multiplication to see if the result is integer.

Example check:

For the option ( x ): $$ x \div 99 $$

If this results in a whole number, ( x ) is divisible by both 9 and 11.

The answer is the number that is divisible by both 9 and 11 from the options given, which is found by checking which one meets the divisibility test.

More Information

The LCM is a useful concept in determining common multiples of integers, particularly when dealing with divisibility. In this case, understanding that both factors must be met simplifies the process to just checking against their LCM.

Tips

Forgetting to calculate the LCM correctly, which can lead to checking against the wrong number.
Skipping the long division and assuming a number is divisible without checking properly.

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