Difficulty: Medium
Correct Answer: 135
Explanation:
Introduction:
Finding the highest common factor (HCF), also known as greatest common divisor (GCD), of two numbers is a core arithmetic skill used in many aptitude questions. This problem asks you to determine the HCF of 6345 and 2160 using systematic factorization or the Euclidean algorithm.
Given Data / Assumptions:
Concept / Approach:
The most efficient method is the Euclidean algorithm, which uses repeated division to find the greatest common divisor. The idea is:
HCF(a, b) = HCF(b, a mod b)and this is repeated until the remainder becomes zero. The last non-zero remainder is the HCF.
Step-by-Step Solution:
Step 1: Arrange numbers.Take a = 6345 and b = 2160.Step 2: Apply Euclidean algorithm.6345 ÷ 2160 gives quotient 2 and remainder r1.2 * 2160 = 4320, so r1 = 6345 − 4320 = 2025.Now HCF(6345, 2160) = HCF(2160, 2025).Step 3: Continue with 2160 and 2025.2160 − 2025 = 135, so remainder r2 = 135.Now HCF(2160, 2025) = HCF(2025, 135).Step 4: Divide 2025 by 135.2025 ÷ 135 = 15 exactly, remainder 0.Therefore, HCF = 135
Verification / Alternative check:
Check that 135 divides both numbers exactly. 6345 ÷ 135 = 47, which is an integer. 2160 ÷ 135 = 16, also an integer. Thus 135 is a common factor. Any greater common divisor would need to be larger than 135, but 135 already embodies all common prime factors, so it is indeed the highest common factor.
Why Other Options Are Wrong:
45: Although 45 divides both numbers, it is not the largest such factor since 135 is bigger and also divides both.
270: 2160 ÷ 270 = 8, but 6345 ÷ 270 is not an integer.
15 and 90: Both divide the numbers but are smaller than 135, so they cannot be the HCF.
Common Pitfalls:
Students may stop at a smaller common factor like 45 or 90, forgetting to check if there is a larger one. Using the Euclidean algorithm avoids guessing and ensures you reach the greatest common factor systematically.
Final Answer:
The highest common factor of 6345 and 2160 is 135.
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