Four metal rods have lengths 78 cm, 104 cm, 117 cm, and 169 cm. They must be cut into equal-length pieces with no wastage. What is the maximum number of pieces that can be obtained in total?

Introduction:
In problems about cutting objects into equal parts without wastage, the key idea is to use the highest common factor, also called greatest common divisor, of the given lengths. This ensures each piece length divides every original length exactly, creating the maximum possible number of pieces.

Given Data / Assumptions:

• Rod lengths: 78 cm, 104 cm, 117 cm, 169 cm
• Pieces must have equal length
• No material wastage is allowed

Concept / Approach:
The longest feasible piece length is HCF(78, 104, 117, 169). Once the piece length is known, the total number of pieces equals the sum of (length / piece length) over all rods.

Step-by-Step Solution:

Verification / Alternative check:
If we chose any piece length greater than 13, at least one rod length would not be divisible, causing wastage or inequality. Therefore 13 cm is maximal.

Why Other Options Are Wrong:
27, 43, 48, or 26 do not match the sum of quotients when using the correct HCF, nor do they arise from any valid common divisor analysis.

Common Pitfalls:
Confusing HCF with LCM, or summing individual HCFs pairwise instead of computing the HCF of all lengths together.

Final Answer:
36