Consecutive odd integers — product given The product of two consecutive odd numbers is 6723. Determine the greater of the two odd integers.

Introduction / Context:
This question checks understanding of consecutive odd numbers and converting a word statement into a simple quadratic equation. Consecutive odd integers differ by 2, and their product is supplied. The task is to retrieve the larger integer directly and verify it.

Given Data / Assumptions:

• The two odd integers are consecutive: let them be n and n + 2.
• Their product equals 6723: n(n + 2) = 6723.
• We seek the greater integer (n + 2).

Concept / Approach:
Form a quadratic equation from the product relation and solve it using the discriminant method. Because the numbers are integers and consecutive odds, the quadratic should factor nicely or yield an integer root after applying the square root to the discriminant.

Step-by-Step Solution:
Start: n(n + 2) = 6723 → n^2 + 2n − 6723 = 0.Compute discriminant: D = 2^2 + 4*6723 = 4 + 26892 = 26896.Recognize 26896 = 164^2, so roots are integers.n = (−2 + 164)/2 = 81 (take positive root).Consecutive odd numbers: 81 and 83 → greater = 83.

Verification / Alternative check:
Multiply 81 * 83 = 81*(80 + 3) = 6480 + 243 = 6723, which matches the given product. Therefore, 83 is correct.

Why Other Options Are Wrong:
89, 85, and 91 are not paired with a neighboring odd to give 6723; 81 is the smaller integer, not the greater one requested.

Common Pitfalls:
Using (n, n + 1) instead of odd spacing of 2; arithmetic errors in discriminant; forgetting to extract the greater value after solving for n.

Final Answer:
83