Find the highest common factor (HCF) of the two numbers 3341 and 3328.

Introduction:
This problem tests your ability to use the Euclidean algorithm to find the highest common factor (HCF) of two numbers that are close in value. The Euclidean method is often faster and more reliable than full prime factorisation for large numbers.

Given Data / Assumptions:

• Two numbers: 3341 and 3328
• We must find their HCF.

Concept / Approach:
The Euclidean algorithm states that HCF(a, b) = HCF(b, a − b) when a is greater than b. Repeatedly applying this rule with smaller and smaller pairs eventually leads to a remainder of 0, and the last non zero remainder is the HCF. This is especially convenient when the two numbers are relatively close together.

Step-by-Step Solution:
Let a = 3341 and b = 3328, with a greater than b. First compute the difference: a − b = 3341 − 3328 = 13 So HCF(3341, 3328) = HCF(3328, 13) Now divide 3328 by 13: 3328 ÷ 13 = 256 exactly (13 × 256 = 3328) Since the remainder is 0, the divisor 13 is the HCF. Therefore, HCF(3341, 3328) = 13

Verification / Alternative check:
We can also check that 3341 is divisible by 13: 13 × 257 = 3341. Both numbers 3341 and 3328 are thus multiples of 13, and no larger number divides both because their difference is 13, and any common divisor must divide this difference as well. Hence 13 is indeed the greatest common divisor.

Why Other Options Are Wrong:
31, 257, and 337 do not divide both 3341 and 3328 exactly. In particular, 257 divides 3341 but not 3328, and 337 is larger than the difference between the numbers, so it cannot be a common divisor. Hence they cannot be the HCF.

Common Pitfalls:
A common mistake is to attempt factorisation of each number separately, which is time consuming and prone to errors. Another mistake is to overlook the simple use of the difference a − b when the numbers are close. Using the Euclidean algorithm streamlines the process significantly.

Final Answer:
The highest common factor of 3341 and 3328 is 13.