What is the common factor of 9 and 12
The Correct Answer and Explanation is:
The common factors of 9 and 12 are the numbers that divide both 9 and 12 without leaving a remainder. To find the common factors, we need to first determine the factors of each number.
Factors of 9:
- The factors of 9 are 1, 3, and 9 because 9 can be divided evenly by these numbers.
Factors of 12:
- The factors of 12 are 1, 2, 3, 4, 6, and 12 because 12 can be divided evenly by these numbers.
Next, we look for the numbers that appear in both lists of factors. These common numbers are:
- 1 and 3
Therefore, the common factors of 9 and 12 are 1 and 3. The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest number that appears in both lists, which is 3.
Explanation:
To better understand why 3 is the greatest common factor, we can consider the process of finding the greatest common divisor (GCD) using prime factorization:
- Prime factorization of 9:
- 9 is divisible by 3, and 3 is a prime number.
- So, the prime factorization of 9 is 9=329 = 3^29=32.
- Prime factorization of 12:
- 12 is divisible by 2 and 3. When you divide 12 by 2, you get 6, and dividing 6 by 3 gives you 2.
- So, the prime factorization of 12 is 12=22×312 = 2^2 \times 312=22×3.
Now, when we compare the prime factorizations of 9 and 12, the only common prime factor is 3, and the smallest power of 3 that appears in both is 313^131, which gives the GCD as 3.
Thus, the greatest common factor of 9 and 12 is 3.
