Three containers hold mixtures measured as 403 kg, 434 kg, and 465 kg. Find the greatest measuring capacity (in kg) that can exactly measure each quantity.

Introduction / Context:
This is a highest common factor (HCF) or greatest common divisor (GCD) question. The largest measure that fits exactly into each total is the GCD of the three weights.

Given Data / Assumptions:

• Quantities: 403 kg, 434 kg, 465 kg
• We seek the greatest capacity dividing all three exactly

Concept / Approach:
Compute the GCD via prime factorization. The common prime factors with the smallest exponents across all numbers define the GCD.

Step-by-Step Solution:
403 = 13 * 31434 = 2 * 217 = 2 * 7 * 31465 = 5 * 93 = 5 * 3 * 31Common factor across all three = 31 ⇒ GCD = 31 kg.

Verification / Alternative check:
403/31 = 13, 434/31 = 14, 465/31 = 15, all integers; hence 31 kg measures each exactly.

Why Other Options Are Wrong:
1 kg is trivial but not the greatest. 7 kg and 13 kg do not divide all three. 41 kg divides none of them exactly.

Common Pitfalls:
Arithmetic mistakes in factorization; stopping at a smaller common factor and missing the larger prime 31.

Final Answer:
31 kg