A general wants to form the largest possible square array from 16160 soldiers but finds 31 soldiers left over. Find the number of soldiers in the front row (the side of the square).

Introduction:
Arranging soldiers in a square means using the greatest perfect square not exceeding the total. Any leftovers indicate how far the count is from a perfect square.

Given Data / Assumptions:

• Total soldiers = 16160
• Left over = 31
• Front row size equals the square root of the used perfect square

Concept / Approach:
If 31 are left, then the number used is 16160 − 31 = 16129. If 16129 is a perfect square, its square root gives the front row size.

Step-by-Step Solution:
Used soldiers = 16160 − 31 = 16129Check: 127^2 = 16129Therefore, side of the square (front row) = 127

Verification / Alternative check:
Compute (126)^2 = 15876 and (128)^2 = 16384 to see that 16129 sits exactly at 127^2.

Why Other Options Are Wrong:
125, 126, 128: Their squares are not 16129.

Common Pitfalls:
Forgetting to subtract the leftover before taking the square root; confusing nearest higher vs lower perfect square.

Final Answer:
127