Question

b. the gcf of any two odd numbers is always odd.

Explanation:

To determine if the GCF of any two odd numbers is always odd, we analyze the properties of odd numbers and GCF.

Step 1: Recall the definition of odd numbers

An odd number can be represented as \( 2k + 1 \) where \( k \) is an integer. Let the two odd numbers be \( m = 2a + 1 \) and \( n = 2b + 1 \), where \( a \) and \( b \) are integers.

Step 2: Recall the definition of GCF

The GCF (Greatest Common Factor) of two numbers is the largest number that divides both of them without leaving a remainder.

Step 3: Analyze the factors of odd numbers

Odd numbers are not divisible by 2. So, any factor of an odd number must also be odd (because if a number had an even factor, the number itself would be even).

Step 4: Consider the GCF of two odd numbers

The GCF of two numbers is a factor of both numbers. Since both numbers are odd, all their factors are odd. Therefore, the largest factor common to both (the GCF) must also be odd.

For example, take two odd numbers: 9 (factors: 1, 3, 9) and 15 (factors: 1, 3, 5, 15). The GCF of 9 and 15 is 3, which is odd. Another example: 21 (factors: 1, 3, 7, 21) and 27 (factors: 1, 3, 9, 27). The GCF is 3, which is odd. Even if we take two odd numbers like 1 and 3, the GCF is 1 (odd). If we take 15 and 25, factors of 15 are 1, 3, 5, 15; factors of 25 are 1, 5, 25. GCF is 5 (odd).

Answer:

The statement "The GCF of any two odd numbers is always odd" is True.