The product of two consecutive odd integers is 6723. What is the greater of the two odd integers?

Introduction / Context:Products of consecutive odd integers often relate to near-square identities. Recognizing patterns like (n - 1)(n + 1) = n^2 - 1 speeds up solving and avoids trial-and-error across many pairs.

Given Data / Assumptions:

• We need odd integers k and k + 2.
• (k)(k + 2) = 6723.
• Integers are positive (reasonable from context).

Concept / Approach:Use the identity k(k + 2) = (k + 1)^2 - 1. Set (k + 1)^2 ≈ 6724, which hints that k + 1 is close to sqrt(6724). If (k + 1)^2 = 6724 exactly, then (k)(k + 2) = 6723, matching the target product.

Step-by-Step Solution:

Verification / Alternative check:Direct multiplication: 81 * 83 = (80 + 1)(80 + 3) = 80^2 + 4*80 + 3 = 6400 + 320 + 3 = 6723.

Why Other Options Are Wrong:85, 89, and 91 do not pair with an adjacent odd to produce 6723. 81 is the smaller of the correct pair, not the greater.

Common Pitfalls:Trying many odd pairs blindly instead of using the near-square trick.

Final Answer:83