Four copper rods have lengths 52 m, 65 m, 78 m, and 91 m. They are to be cut into equal-length pieces with no wastage. What is the minimum total number of pieces obtained?

Introduction:
To cut rods into equal parts without any waste, we choose the piece length as the highest common factor of all given lengths. This maximizes piece length and therefore minimizes the total number of pieces.

Given Data / Assumptions:

• Rod lengths: 52 m, 65 m, 78 m, 91 m
• Equal piece length
• No wastage

Concept / Approach:
Piece length = HCF(52, 65, 78, 91). Then number of pieces = sum of (length / HCF).

Step-by-Step Solution:

Verification / Alternative check:
Any greater common length does not exist, so 13 m is optimal, giving the minimum number of pieces.

Why Other Options Are Wrong:
20, 21, 23, and 24 do not match the arithmetic based on the correct HCF and quotients.

Common Pitfalls:
Using LCM instead of HCF or adding lengths before dividing. Always divide each length by the HCF and sum the quotients.

Final Answer:
22