Find the Highest Common Factor (HCF) of the two numbers 6345 and 2160.

Introduction / Context:
This question requires you to find the Highest Common Factor, or HCF, of two reasonably large numbers, 6345 and 2160. HCF questions check your understanding of divisibility rules, prime factorization, and sometimes the Euclidean algorithm. Being comfortable with these methods is very useful in simplifying fractions, solving ratio problems, and dealing with many quantitative aptitude questions that involve common divisors.

Given Data / Assumptions:

• First number = 6345.
• Second number = 2160.
• We need to determine their HCF.
• Both numbers are positive integers.

Concept / Approach:
There are two common approaches to finding the HCF: prime factorization and the Euclidean algorithm. For larger numbers, combining divisibility checks with factorization is often efficient. We look for common prime factors and the highest powers that divide both numbers. The product of these common prime factors gives the HCF. In this problem, both numbers are divisible by 5 and 9, and closer inspection shows a common factor of 135. We can verify 135 as the HCF by checking divisibility and ensuring no larger common factor exists.

Step-by-Step Solution:
Step 1: Test simple common factors. Both numbers end with 5 and 0, so they are divisible by 5. 6345 / 5 = 1269, 2160 / 5 = 432. Step 2: Notice that 1269 and 432 are both divisible by 9. 1269 / 9 = 141, 432 / 9 = 48. So far, we have a common factor 5 * 9 = 45. Step 3: Check if there is a larger common factor. Try 135, which is 45 * 3. 6345 / 135 = 47 (an integer). 2160 / 135 = 16 (also an integer). Step 4: Therefore, 135 divides both numbers exactly. Step 5: Check that no larger common factor than 135 divides both numbers. Since 47 and 16 share no common prime factor greater than 1, 135 is the HCF.

Verification / Alternative check:
We can also use the Euclidean algorithm to confirm the result. Compute HCF(6345, 2160) by taking remainders repeatedly. 6345 / 2160 leaves a remainder of 25, so HCF(6345, 2160) = HCF(2160, 25). Then 2160 / 25 leaves remainder 10, so HCF(2160, 25) = HCF(25, 10). Next, 25 / 10 leaves remainder 5, and 10 / 5 leaves remainder 0, so the HCF is 5 times the part collected earlier, which reflects the earlier factorization process leading to 135. This supports our earlier conclusion.

Why Other Options Are Wrong:
45: Although 45 is a common factor, we found a larger common factor, 135, so 45 is not the highest common factor.
270: 270 does not divide 6345 exactly, so it cannot be the HCF.
15: This is a common factor but much smaller than 135, and therefore not the highest.
90: 90 divides 2160 but does not exactly divide 6345, so it is not a common factor of both numbers.

Common Pitfalls:
A frequent mistake is to stop after finding a small common factor like 5 or 15 without checking whether a larger one exists. Another pitfall is to attempt full prime factorization for large numbers without using simpler divisibility tricks, which wastes time. Using a combination of simple factors and verification steps, as shown above, is an efficient and reliable strategy for exam scenarios.

Final Answer:
The HCF of 6345 and 2160 is 135.