Greatest Common Factor of 21 and 36

Method 1: Listing the factors

To find the factors of 21, we can list all the numbers that divide it evenly:

1, 3, 7, 21

To find the factors of 36, we can list all the numbers that divide it evenly:

1, 2, 3, 4, 6, 9, 12, 18, 36

Now we look for the largest number that appears in both lists, which is 3. Therefore, the GCF of 21 and 36 is 3.

Method 2: Prime factorization

To find the prime factorization of 21, we can divide it by its smallest prime factor, which is 3:

21 ÷ 3 = 7

Since 7 is a prime number, we stop. Therefore, the prime factorization of 21 is 3 × 7.

To find the prime factorization of 36, we can divide it by its smallest prime factor, which is 2:

36 ÷ 2 = 18

18 ÷ 2 = 9

9 ÷ 3 = 3

Since 3 is a prime number, we stop. Therefore, the prime factorization of 36 is 2² × 3².

Now we look for the common prime factors and their smallest exponents between the two factorizations, which are 3 (appearing once in 3 × 7) and 2 (appearing twice in 2² × 3²).

To get the GCF, we multiply these common prime factors with their smallest exponents:

GCF = 3 × 2² = 12

Therefore, the GCF of 21 and 36 is 3.

The greatest common factor (GCF) of 21 and 36 is 3.

The greatest common factor (GCF) of 21 and 36 is 3.

There are different methods to find the greatest common factor (GCF) of 21 and 36, such as:

1. Prime factorization method: To find the prime factorization of 21 and 36, we can write them as products of primes:

21 = 3 × 7 36 = 2 × 2 × 3 × 3

Then, we identify the common prime factors between the two numbers, which are 3. We take the product of the common prime factors with the lowest exponent, which is 3, and get the GCF:

GCF(21, 36) = 3

2. Listing the factors method: To find the factors of 21 and 36, we list all the positive integers that divide them evenly:

Factors of 21: 1, 3, 7, 21 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

We identify the common factors between the two numbers, which are 1, 3, and even though 36 has more factors, we only consider the factors that are common to both 21 and 36. We choose the greatest common factor, which is 3:

GCF(21, 36) = 3

Either method works to find the GCF of 21 and 36.

The Euclidean algorithm is another method to find the greatest common factor (GCF) of 21 and 36. It is based on the idea that the GCF of two numbers is the same as the GCF of one of the numbers and the remainder of the division of the other number by the first number. We apply this algorithm as follows:

1. Divide the larger number by the smaller number and find the remainder:

36 ÷ 21 = 1 with a remainder of 15

2. Divide the smaller number (21) by the remainder (15) and find the new remainder:

21 ÷ 15 = 1 with a remainder of 6

3. Divide the remainder (15) by the new remainder (6) and find the new remainder:

15 ÷ 6 = 2 with a remainder of 3

4. Divide the new remainder (6) by the last remainder (3) and find the final remainder:

6 ÷ 3 = 2 with a remainder of 0

Since the last remainder is zero, we stop. The GCF is the last non-zero remainder, which is 3. Therefore:

GCF(21, 36) = 3

This method may take a few more steps than the previous methods, but it is a systematic way to find the GCF using only divisions and remainders.

To find the greatest common factor (GCF) of 21 and 36 using prime factorization, we can write each number as a product of its prime factors:

21 = 3 x 7 36 = 2 x 2 x 3 x 3

Then, we identify the prime factors that are common to both numbers, which are 3. Note that 36 has an extra factor of 2, but since 2 is not a factor of 21, it is not common to both numbers. Therefore, the only common prime factor is 3.

To find the GCF, we multiply the common prime factor, 3, by the smallest power of 3 that appears in either factorization. Since 3 appears once in the factorization of 21 and twice in the factorization of 36, we choose the smaller exponent, which is 1.

Therefore, the GCF of 21 and 36 is:

GCF(21, 36) = 3^1 = 3

Therefore, the GCF of 21 and 36 is 3.

To find the greatest common factor (GCF) of 21 and 36 by listing all common factors, we need to list all the positive integers that divide both 21 and 36 evenly. Then, we choose the greatest common factor from this list.

The factors of 21 are: 1, 3, 7, and 21.

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

To find the common factors of 21 and 36, we need to identify the numbers that appear in both lists. The common factors are: 1, 3.

Therefore, the GCF of 21 and 36 is the greatest common factor among these common factors, which is 3.

Hence, we can conclude that:

GCF(21, 36) = 3

Here are some frequently asked questions (FAQ) about the greatest common factor (GCF) of 21 and 36:

Q: What is the GCF of 21 and 36? A: The GCF of 21 and 36 is 3.

Q: What are the methods to find the GCF of 21 and 36? A: There are several methods to find the GCF of 21 and 36, including prime factorization, listing all common factors, and the Euclidean algorithm.

Q: What is prime factorization? A: Prime factorization is a method of expressing a positive integer as a product of its prime factors.

Q: What are the prime factors of 21 and 36? A: The prime factors of 21 are 3 and 7. The prime factors of 36 are 2 and 3.

Q: What is the Euclidean algorithm? A: The Euclidean algorithm is a method of finding the GCF of two numbers by repeatedly dividing the larger number by the smaller number and finding the remainder until the remainder is zero.

Q: How do I know which method to use to find the GCF of 21 and 36? A: It depends on your preference and familiarity with the different methods. However, all the methods should give you the same answer. You can choose the method that you find easiest or most convenient.

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