Set A consists of consecutive integers. What is the median of all the numbers in set A? The smallest number...

Understanding the Question

We need to find the median of set A, which consists of consecutive integers.

What We Need to Determine

To find the median of consecutive integers, we need to identify:

• The middle value (if we have an odd number of integers)
• The average of the two middle values (if we have an even number of integers)

Key Insight

For consecutive integers, the median equals the mean. But to find either one, we need to know exactly which consecutive integers we're dealing with.

What would make this sufficient? We need enough information to determine the complete set of consecutive integers.

Analyzing Statement 1

Statement 1 tells us: The smallest number in set A is 4.

What This Gives Us

We know the set starts at 4, so we have \{4, 5, 6, ...\}, but we don't know where it ends.

Testing Different Scenarios

Let's see what happens with different ending points:

• If \mathrm{A} = \{4, 5, 6\} (3 integers) → \mathrm{median} = 5
• If \mathrm{A} = \{4, 5, 6, 7\} (4 integers) → \mathrm{median} = \frac{5 + 6}{2} = 5.5
• If \mathrm{A} = \{4, 5, 6, 7, 8\} (5 integers) → \mathrm{median} = 6

Different counts give us different medians.

Conclusion

Since we can't determine a unique median, Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Important: We now forget Statement 1 completely and analyze Statement 2 by itself.

Statement 2 tells us: The standard deviation of all numbers in set A is \sqrt{2}.

Key Pattern Recognition

Here's a crucial insight: For consecutive integers, the standard deviation depends only on how many integers there are, not their actual values. Each count of consecutive integers has its own unique standard deviation.

This means that a standard deviation of \sqrt{2} tells us we have exactly one specific count of consecutive integers. (Through the mathematical relationship between spread and count, this corresponds to exactly 5 consecutive integers.)

What We Still Don't Know

While we now know there are 5 consecutive integers, we don't know which 5:

• Could be \{1, 2, 3, 4, 5\} → \mathrm{median} = 3
• Could be \{10, 11, 12, 13, 14\} → \mathrm{median} = 12
• Could be \{100, 101, 102, 103, 104\} → \mathrm{median} = 102

Same count, different starting points, different medians.

Conclusion

Since we can't determine a unique median, Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] We've now eliminated choices A, B, and D.

Combining Statements

Now let's see what happens when we use both statements together.

Combined Information

• From Statement 1: The set starts at 4
• From Statement 2: The set contains exactly 5 consecutive integers

The Complete Picture

Now we can determine the complete set: \mathrm{A} = \{4, 5, 6, 7, 8\}

With 5 consecutive integers starting at 4, we have:

• The median is the middle value (3rd position)
• Therefore, \mathrm{median} = 6

Why This Works

Together, the statements give us:

• The starting point (where to begin)
• The count (how many to include)

This uniquely determines the entire set, and therefore its median.

[STOP - Sufficient!] The statements together are sufficient.

The Answer: C

Both statements together give us exactly what we need, but neither alone is enough.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."