P, Q, R, S, T and U are six consecutive odd numbers. Their average is 52. What is the product P × U, where P is the smallest of the six numbers and U is the largest?

Introduction / Context:
This problem deals with consecutive odd numbers and their average. When numbers are evenly spaced in an arithmetic progression, the average has a special relationship with the middle terms. Here you are asked to use that relationship to determine the exact values of the smallest and largest numbers and then compute their product.

Given Data / Assumptions:

• P, Q, R, S, T and U are six consecutive odd integers.

• Their average is 52.

• The numbers are in increasing order from P to U.

• We must find the product P × U.

Concept / Approach:
Six consecutive odd numbers form an arithmetic progression with common difference 2. A convenient way is to express them in terms of a central value. For six consecutive odd numbers, we can write them as n − 5, n − 3, n − 1, n + 1, n + 3 and n + 5, where n is the average. Since the average is given as 52, that helps locate all six numbers. Once we know P and U explicitly, we simply multiply them to get their product.

Step-by-Step Solution:
Let the six consecutive odd numbers be n − 5, n − 3, n − 1, n + 1, n + 3, n + 5. The average of an arithmetic progression equals its central value, so average = n. Given average = 52, therefore n = 52. So the numbers are 52 − 5 = 47, 52 − 3 = 49, 52 − 1 = 51, 52 + 1 = 53, 52 + 3 = 55 and 52 + 5 = 57. Hence P = 47 (smallest) and U = 57 (largest). Product P × U = 47 * 57. Compute: 47 * 57 = 47 * (50 + 7) = 47 * 50 + 47 * 7 = 2350 + 329 = 2679.

Verification / Alternative check:
We can verify the average. Sum of the numbers = 47 + 49 + 51 + 53 + 55 + 57. Pairing symmetric numbers: (47 + 57) + (49 + 55) + (51 + 53) = 104 + 104 + 104 = 312. Average = 312 / 6 = 52, which matches the given average. This confirms that the set of numbers is correct and therefore the product 2679 is reliable.

Why Other Options Are Wrong:
Other products such as 2212, 2154 or 2349 would correspond to incorrect pairs of numbers or miscalculations. For example, even a small change in one factor, such as using 45 instead of 47, would drastically alter the product. Since the average fixes the sequence precisely, there is only one possible pair P and U, and only 2679 matches their product.

Common Pitfalls:
One common issue is to assume that 52 itself must be one of the odd numbers, but 52 is even, so it acts only as the central average, not as a term. Students also sometimes mis assign the offsets and might use n − 4 or n − 2, which would give even numbers. Always ensure the step size between consecutive odd numbers is 2 and that all resulting values remain odd integers.

Final Answer:
The product of P and U is 2679.