Difficulty: Medium
Correct Answer: 195 litres
Explanation:
Introduction / Context:
This problem is a direct application of the highest common factor (HCF) in a measurement context. We want one measuring vessel that will exactly measure three given quantities of liquid without leaving any remainder. In such cases, the largest possible capacity of the vessel is equal to the HCF of the three quantities.
Given Data / Assumptions:
Concept / Approach:
The largest measure that can be used with all three containers is their HCF. This is because each capacity must be an exact multiple of the measure. To find the HCF, we perform prime factorization or use the Euclidean algorithm. Since the numbers are moderately large, prime factorization combined with divisibility checks is very helpful.
Step-by-Step Solution:
First, find the HCF of 1365 and 1560.
1365 factors: 1365 = 3 * 5 * 7 * 13.
1560 factors: 1560 = 2^3 * 3 * 5 * 13.
Common factors so far: 3 * 5 * 13 = 195.
Now check 1755.
1755 = 3 * 5 * 3 * 3 * 13 = 3^3 * 5 * 13.
1755 is divisible by 195 because 1755 / 195 = 9.
Therefore, HCF(1365, 1560, 1755) = 195 litres.
Verification / Alternative check:
We can verify:
1365 / 195 = 7, an integer.
1560 / 195 = 8, an integer.
1755 / 195 = 9, an integer.
No option larger than 195 divides all three values exactly, so 195 litres is the largest possible measure that satisfies the requirement.
Why Other Options Are Wrong:
37, 97, 129, and 65 either do not divide all three capacities exactly or are smaller than the maximum common factor 195, so they cannot be the largest possible common measure.
Common Pitfalls:
One mistake is to confuse HCF with LCM, which would give a very large number that is not a measure but a combined multiple. Another error is to check divisibility only for one or two containers and overlook the third. It is essential to ensure that the chosen measure divides all three quantities without remainder.
Final Answer:
195 litres
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