Question

How many integers are there between m and n, exclusive, if m and n are themselves integers?
(1) m - n = 8
(2) There are 5 integers between, but not including, m - 1 and n - 1.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

The number of integers strictly between two integers `a` and `b` is `|a - b| - 1`. The number of integers between `a-k` and `b-k` is the same as the number of integers between `a` and `b`. Statement (2) effectively tells you the answer is 5.

Answer 1

The Correct Option is D

Step 1: Understanding the Question
The question asks for the number of integers strictly between m and n. The formula for the number of integers between two integers m and n (exclusive) is \(|m - n| - 1\). We need to find the value of this expression.

Step 2: Analysis of Statement (1)
Statement (1) gives us \(m - n = 8\). This tells us the difference between m and n is 8. Using the formula, the number of integers between them is: \[ |m - n| - 1 = |8| - 1 = 8 - 1 = 7 \] For example, if m=10 and n=2, then m-n=8. The integers between them are 3, 4, 5, 6, 7, 8, 9, which is a total of 7 integers. The result is always 7. This provides a unique answer.
Therefore, Statement (1) ALONE is sufficient.

Step 3: Analysis of Statement (2)
Statement (2) says there are 5 integers between m-1 and n-1. Let's apply the same formula to these new endpoints. The number of integers between m-1 and n-1 is: \[ |(m-1) - (n-1)| - 1 \] We are told this value is 5. \[ |(m-1) - (n-1)| - 1 = 5 \] \[ |m - 1 - n + 1| - 1 = 5 \] \[ |m - n| - 1 = 5 \] \[ |m - n| = 6 \] Now we can use this result to answer the original question. The number of integers between m and n is: \[ |m - n| - 1 = 6 - 1 = 5 \] This also provides a unique answer.
Therefore, Statement (2) ALONE is sufficient.
Step 4: Final Answer
Since each statement alone is sufficient to answer the question, the correct answer is (D).