In a locker room each row of lockers has the same number of lockers, and the number of rows is...

Understanding the Question

Let's break down what we're dealing with in this locker room setup:

• Every row has exactly the same number of lockers
• The number of rows equals exactly half the number of lockers in each row

If we call the number of lockers in a row "L", then we have L/2 rows. This creates a special relationship:
Total lockers = \(\mathrm{L} \times (\mathrm{L}/2) = \mathrm{L}^2/2\)

What we need to find: The exact value of L (number of lockers in a row).

Key insight: Since the total must equal \(\mathrm{L}^2/2\), this means the total number of lockers must be half of a perfect square. This pattern recognition will guide our entire analysis.

Analyzing Statement 1

Statement 1: There are 72 lockers total in the locker room.

Using our insight, we know that total = \(\mathrm{L}^2/2\), so:

• \(\mathrm{L}^2/2 = 72\)
• Therefore, \(\mathrm{L}^2 = 144\)

Now we need to find which perfect square equals 144. Since \(12 \times 12 = 144\), we know L = 12.

Quick verification:

• 12 lockers per row
• \(12/2 = 6\) rows
• Total: \(12 \times 6 = 72\) ✓

This gives us exactly one value for L.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The number of rows is 6 less than the number of lockers in a row.

From the question setup, we know:

• Number of rows = \(\mathrm{L}/2\)

Statement 2 tells us:

• Number of rows = L - 6

Since both expressions equal the number of rows:
\(\mathrm{L}/2 = \mathrm{L} - 6\)

Let's think strategically: We need a number where half of it equals the number minus 6.

Testing L = 12:

• Half of 12 = 6
• 12 minus 6 = 6
• These match! ✓

Is this the only value? Yes - you can verify that L = 12 is the unique solution to this relationship.

[STOP - Statement 2 is SUFFICIENT!]

This eliminates choice C (and confirms our answer isn't A).

The Answer

Both statements independently lead us to the exact same answer: 12 lockers per row.

• Statement 1 alone → L = 12
• Statement 2 alone → L = 12

Since each statement alone is sufficient to answer the question:

Answer: D - Each statement alone is sufficient.