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In a locker room each row of lockers has the same number of lockers, and the number of rows is \(\frac{1}{2}\) the number of lockers in a row. How many lockers are in a row?
Let's break down what we're dealing with in this locker room setup:
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.
Statement 1: There are 72 lockers total in the locker room.
Using our insight, we know that total = \(\mathrm{L}^2/2\), so:
Now we need to find which perfect square equals 144. Since \(12 \times 12 = 144\), we know L = 12.
Quick verification:
This gives us exactly one value for L.
[STOP - Statement 1 is SUFFICIENT!]
This eliminates choices B, C, and E.
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:
Statement 2 tells us:
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:
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).
Both statements independently lead us to the exact same answer: 12 lockers per row.
Since each statement alone is sufficient to answer the question:
Answer: D - Each statement alone is sufficient.