Determine the highest common factor (HCF) of the three positive integers 1, 2, and 3, and explain why this value is the greatest integer that divides all three numbers exactly.

Introduction / Context:
This question checks a basic but very important concept in number theory, the highest common factor (HCF) of a small set of integers. Even though the numbers are simple, understanding why the HCF is what it is lays the foundation for more complex problems involving divisibility, fractions, and simplifying ratios.

Given Data / Assumptions:

• Numbers involved: 1, 2, and 3
• All are positive integers
• We are looking for the greatest integer that divides each of them exactly.

Concept / Approach:
The HCF of a set of integers is the largest positive integer that divides every number in the set without leaving any remainder. For small numbers, we can list the divisors and find their intersection. For larger numbers, we often use prime factorization or the Euclidean algorithm. For this very small set, simple listing is enough.

Step-by-Step Solution:
Divisors of 1: 1 Divisors of 2: 1, 2 Divisors of 3: 1, 3 Common divisors of 1, 2, and 3: only 1. Therefore, the greatest common divisor or HCF is 1.

Verification / Alternative check:
Confirm divisibility: 1 divided by 1 gives 1. 2 divided by 1 gives 2. 3 divided by 1 gives 3. All results are integers. No integer greater than 1 divides 1, so there cannot be any higher common factor than 1.

Why Other Options Are Wrong:
2 and 3 do not divide 1, so they cannot be common factors of the entire set. 0 is not considered as an HCF in standard number theory, and 6 is larger than any of the given numbers and does not divide them all.

Common Pitfalls:
A typical misunderstanding is to think that the HCF must be one of the larger given numbers or that 0 can be a highest common factor. Remember that the HCF must divide every number exactly and must be positive. With 1 in the list, the HCF can never exceed 1.

Final Answer:
1