Finding Common Ground | FIO

Question 7

Make a general statement about the HCF for the following pairs of numbers. You could consider examples before coming up with general statements. Look for possible explanations of why they hold.

(a) Two consecutive even numbers

(b) Two consecutive odd numbers

(d) Two consecutive numbers

(e) Two co-prime numbers

Share your observations with the class.

Solution

Understand the Question

The Highest Common Factor (HCF) of two numbers is the largest integer that divides both of them without leaving a remainder.
A fundamental property of division is that any common factor of two numbers must also divide their difference.
We can observe patterns through specific numerical examples and then formulate general algebraic proofs using the properties of even, odd, and co-prime numbers.

(a) Two consecutive even numbers

Step 1 · Find HCF of Consecutive Even Numbers

Examples:

For $(2, 4)$ : Factors of $2 = 1, 2$ ; Factors of $4 = 1, 2, 4 \implies \text{HCF} = 2$
For $(6, 8)$ : Factors of $6 = 1, 2, 3, 6$ ; Factors of $8 = 1, 2, 4, 8 \implies \text{HCF} = 2$
For $(10, 12)$ : Factors of $10 = 1, 2, 5, 10$ ; Factors of $12 = 1, 2, 3, 4, 6, 12 \implies \text{HCF} = 2$

General Explanation: Let two consecutive even numbers be $2n$ and $2n + 2$ .

Their difference is:

(2n + 2) - 2n = 2

Any common factor of two numbers must divide their difference. Since the difference is $2$ , the only possible common factors are $1$ and $2$ . Because both numbers are even, $2$ is always a common factor. Therefore, the HCF must be $2$ .

Answer

(a) $2$

(b) Two consecutive odd numbers

Step 1 · Find HCF of Consecutive Odd Numbers

Examples:

For $(3, 5)$ : Common factor is $1 \implies \text{HCF} = 1$
For $(7, 9)$ : Common factor is $1 \implies \text{HCF} = 1$
For $(11, 13)$ : Common factor is $1 \implies \text{HCF} = 1$

General Explanation: Let two consecutive odd numbers be $2n - 1$ and $2n + 1$ .

Their difference is:

(2n + 1) - (2n - 1) = 2

Any common factor must divide their difference, $2$ . The factors of $2$ are $1$ and $2$ . Since both numbers are odd, $2$ cannot be a factor. Therefore, the HCF is always $1$ .

Answer

(b) $1$

(c) Two even numbers

Step 1 · Find HCF of Two Even Numbers

Examples:

For $(4, 10)$ : Common factors are $1, 2 \implies \text{HCF} = 2$
For $(8, 12)$ : Common factors are $1, 2, 4 \implies \text{HCF} = 4$
For $(14, 20)$ : Common factors are $1, 2 \implies \text{HCF} = 2$

General Explanation: Let the two even numbers be $2m$ and $2n$ (where $m, n$ are integers).

Since both numbers are even, $2$ is guaranteed to be a common factor. The HCF is the greatest common factor, which means it must be a multiple of $2$ . Therefore, the HCF is always an even number (at least $2$ ).

Answer

(c) An even number (at least $2$ )

(d) Two consecutive numbers

Step 1 · Find HCF of Consecutive Numbers

Examples:

For $(7, 8)$ : Common factor is $1 \implies \text{HCF} = 1$
For $(14, 15)$ : Common factor is $1 \implies \text{HCF} = 1$
For $(20, 21)$ : Common factor is $1 \implies \text{HCF} = 1$

General Explanation: Let the two consecutive integers be $n$ and $n + 1$ .

Their difference is:

(n + 1) - n = 1

Any common factor must divide their difference $1$ . The only factor of $1$ is $1$ . Therefore, the HCF of any two consecutive numbers is always $1$ .

Answer

(d) $1$

(e) Two co-prime numbers

Step 1 · Find HCF of Co-prime Numbers

Examples:

For $(4, 9)$ : Common factor is $1 \implies \text{HCF} = 1$
For $(5, 8)$ : Common factor is $1 \implies \text{HCF} = 1$

General Explanation: By definition, two numbers are co-prime if they share no common positive factors other than $1$ . Therefore, their Highest Common Factor is always $1$ .

Answer

(e) $1$

Common Mistakes

Consecutive Even vs. Any Even: The HCF of consecutive even numbers is always strictly $2$ , whereas the HCF of any two even numbers can be any even number ( $2, 4, 6, \dots$ ).
Co-prime vs. Prime: Co-prime numbers do not both need to be prime numbers (e.g., $4$ and $9$ are both composite, but they are co-prime because $\text{HCF}(4, 9) = 1$ ).
Difference Property: Forgetting that common factors must divide the difference between two numbers is a missed shortcut for determining HCF constraints.

Question 7

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