Find the Highest Common Factor (HCF) of the two numbers 3341 and 3328.

Introduction / Context:
This problem focuses on finding the Highest Common Factor, or HCF, of two given numbers. The HCF is the largest positive integer that divides both numbers exactly, with no remainder. Questions of this type measure understanding of divisibility, factors, and sometimes the use of simple techniques like taking differences. HCF problems are fundamental in number theory and very important for competitive exams and school level aptitude tests.

Given Data / Assumptions:

• First number = 3341.
• Second number = 3328.
• We need to determine the HCF of these two numbers.
• No additional hidden conditions are present.

Concept / Approach:
When two numbers are close to each other, a very efficient approach to find their HCF is to consider their difference. Any common factor of the two numbers must also divide their difference. Once the difference is found, we test if this difference divides both numbers exactly. If it does, that difference is a strong candidate for the HCF. We can then confirm by checking divisibility and ensuring that no larger common factor exists.

Step-by-Step Solution:
Step 1: Compute the difference between the two numbers. Difference = 3341 - 3328 = 13 Step 2: Check if 13 divides both numbers exactly. 3341 / 13 = 257 (an integer) 3328 / 13 = 256 (an integer) Step 3: Since 13 divides both numbers exactly, it is a common factor. Step 4: Because the numbers are close, and 13 is already a significant prime factor, there is no larger common factor than 13 that divides both. Therefore, HCF(3341, 3328) = 13.

Verification / Alternative check:
An alternative approach is to use the Euclidean algorithm. We repeatedly take remainders until we reach zero. Starting with 3341 and 3328, the difference based method we used is essentially a simplified Euclidean approach. Because we found that the difference is 13 and it divides both numbers exactly, and no larger candidate exists, the result is confirmed. This gives us confidence that 13 is indeed the highest common factor of the two numbers.

Why Other Options Are Wrong:
31: Does not divide both 3341 and 3328 exactly, so it cannot be the HCF.
257: Divides 3341, but 3328 / 257 is not an integer, so it is not a common factor.
337: Divides neither of the numbers exactly, so it is invalid.
1: While 1 is always a common factor, it is not the highest common factor in this case because we have already found 13 as a larger common divisor.

Common Pitfalls:
A typical mistake is to guess factors without checking divisibility carefully or to assume that because numbers are large, the HCF must also be large. Another mistake is ignoring the difference method, which is especially helpful when numbers are close. Some learners also confuse HCF with LCM and may try to multiply factors instead of looking for common divisors. Always remember that HCF is about the greatest number that divides all given numbers exactly.

Final Answer:
The HCF of 3341 and 3328 is 13.