The product of three consecutive odd integers is 1287. What is the largest of these three odd integers?

Introduction / Context:
This question involves consecutive odd integers and their product. We are told that the product of three consecutive odd numbers equals 1287 and asked to determine the largest of the three. It checks pattern recognition, number sense, and sometimes the ability to guess intelligently around cube roots.

Given Data / Assumptions:

- There are three consecutive odd integers.

- Let them be of the form n - 2, n, and n + 2 or n, n + 2, and n + 4.

- Their product is 1287.

- We must find the largest of these three odd integers.

Concept / Approach:
Consecutive odd integers are close together, so their product will be near the cube of the middle number. We can use approximate cube roots of 1287 to locate likely candidates and then test small sets of three consecutive odd numbers around that estimate. Direct trial is quite efficient here because the numbers are modest.

Step-by-Step Solution:
Step 1: Estimate the middle number. The cube root of 1000 is 10 and of 1331 is 11, so the middle number is likely around 11. Step 2: Try 11 as the middle odd integer. Then the three consecutive odd integers would be 9, 11, and 13. Step 3: Compute the product 9 * 11 * 13. 9 * 11 = 99. 99 * 13 = 99 * 10 + 99 * 3 = 990 + 297 = 1287. So 9, 11, and 13 indeed multiply to 1287. Step 4: Identify the largest of these integers. The three integers are 9, 11, and 13, and the largest is 13.

Verification / Alternative check:
To be sure, we can check other nearby sets. For example, 11, 13, 15 have product 11 * 13 * 15 = 2145, which is too large. Similarly, 7, 9, 11 have product 693, too small. So 9, 11, 13 is the unique suitable triple that produces 1287, confirming that 13 is the largest.

Why Other Options Are Wrong:
- 9: This is actually the smallest of the three integers.

- 11: This is the middle integer, not the largest.

- 17: 15, 17, 19 or 13, 15, 17 would produce much larger products than 1287.

Common Pitfalls:
A typical mistake is assuming the numbers are consecutive integers (like 10, 11, 12) instead of consecutive odd integers. Another error is choosing the wrong triple around the estimate or miscomputing the product. Careful multiplication and a logical search around the estimated cube root avoid these problems.

Final Answer:
The largest of the three consecutive odd integers is 13.