Among the integers from 9 to 54 inclusive, how many numbers are exactly divisible by 9 but not divisible by 3?

Introduction / Context:
Divisibility questions test your understanding of basic number properties and relationships between factors. This problem asks you to examine numbers in a specified range and apply divisibility conditions involving the numbers 9 and 3. Recognising the relationship between divisibility by 9 and divisibility by 3 is the key insight here.

Given Data / Assumptions:
- We are considering all integers from 9 to 54 inclusive. - We want numbers that are divisible by 9. - Among those, we must retain only numbers that are not divisible by 3. - Standard divisibility rules for 3 and 9 apply.

Concept / Approach:
The crucial concept is that if a number is divisible by 9, it is always divisible by 3, because 9 itself is 3 multiplied by 3. In other words, divisibility by 9 implies divisibility by 3. Therefore, a number cannot simultaneously be divisible by 9 and not divisible by 3. Understanding this theoretical relationship means we do not actually need to list or test individual numbers in the range, because the condition is logically impossible.

Step-by-Step Solution:
Step 1: Recall that if a number is divisible by 9, it can be written as 9 * k for some integer k. Step 2: Since 9 = 3 * 3, 9 * k = 3 * (3 * k). Step 3: This shows that any number of the form 9 * k is also a multiple of 3, and hence divisible by 3. Step 4: The question asks for numbers that are divisible by 9 but not by 3. Step 5: From the previous steps, this is impossible because divisibility by 9 always includes divisibility by 3. Step 6: Therefore, within any range, including 9 to 54, no integer can satisfy the condition "divisible by 9 but not divisible by 3." Step 7: Hence, the required count of such numbers is zero.

Verification / Alternative check:
As a quick check, you can list the multiples of 9 between 9 and 54 inclusive: 9, 18, 27, 36, 45, 54. Each of these has a digit sum that is a multiple of 9 and therefore a multiple of 3, confirming that they are all divisible by 3 as well. Since every candidate multiple of 9 fails the "not divisible by 3" condition, the count remains zero. This practical verification matches the theoretical reasoning and confirms the answer.

Why Other Options Are Wrong:
5: There are six multiples of 9 in the given range, not five, and all of them are divisible by 3. 6: This is the number of multiples of 9 but does not account for the divisibility by 3 condition. 9: This overestimates the number of multiples of 9 and still ignores the impossibility implied by the divisibility rules.

Common Pitfalls:
Some students misinterpret the condition and only count multiples of 9, forgetting to apply the additional restriction about 3. Others may mistakenly think that divisibility by 9 and divisibility by 3 are independent, when in fact divisibility by 9 is a stronger condition that automatically includes divisibility by 3. Remembering the hierarchy of factors helps to avoid this conceptual mistake.

Final Answer:
There are 0 numbers between 9 and 54 that are divisible by 9 but not divisible by 3.