Consecutive odd integers — The product of two consecutive odd numbers is 19043. Determine the smaller odd integer, using a neat square-approximation check.

Introduction / Context:Products of consecutive odd integers can be solved either by forming a quadratic or by using a square-approximation strategy. Since consecutive odd numbers are centered around their average, comparing the target product to a nearby perfect square quickly leads to the correct pair without heavy computation.

Given Data / Assumptions:

• Let the two consecutive odd numbers be x and x + 2.
• Product: x(x + 2) = 19043.
• We seek the smaller integer x.

Concept / Approach:Form the quadratic: x^2 + 2x - 19043 = 0. Alternatively, note that for consecutive odd numbers, the product is close to the square of the middle number (x + 1). Compute sqrt(19043) to locate the vicinity and test the nearest odd pair. This avoids full quadratic solving in most test settings.

Step-by-Step Solution:Estimate sqrt(19043) ≈ 138 (since 138^2 = 19044).Consecutive odd numbers around this are 137 and 139.Compute 137 * 139 = 19043.Therefore, the smaller odd integer is 137.

Verification / Alternative check:Quadratic method: x^2 + 2x - 19043 = 0 has integer roots at x = 137 and x = -139 (discard negative for this context). The relevant consecutive odd pair is (137, 139).

Why Other Options Are Wrong:

• 131 / 133 / 129 / 135: Their paired products with the next odd number are far from 19043 and do not match the target.

Common Pitfalls:Testing even numbers; forgetting that the numbers differ by 2; approximating the square root too loosely and not checking the nearest odd pair.

Final Answer:137