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Introduction:

A common factor is a term used in mathematics to describe a number or variable that divides two or more integers exactly. In other words, it is a number that is a factor of two or more other numbers. For example, the number 2 is a common factor of 4 and 6 because it divides both numbers exactly.

Common factors play an important role in various mathematical operations and concepts. They are used in finding the greatest common factor (GCF) of two or more numbers, which is the largest number that divides all the given numbers evenly. The GCF is often used in simplifying fractions and solving problems related to ratios and proportions.

To find the common factors of two or more numbers, we need to identify the factors of each number and then look for the ones that are common to all of them. For example, let’s find the common factors of 12, 18, and 24. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The common factors of all three numbers are 1, 2, 3, 6, and 12.

When finding the GCF, we need to identify the highest common factor among the common factors. In this case, the highest common factor is 12, so the GCF of 12, 18, and 24 is 12.

Common factors are also used in simplifying algebraic expressions. When two or more terms have a common factor, we can factor out that common factor and simplify the expression. For example, consider the expression 6x + 12y. Both terms have a common factor of 6, so we can factor out 6 to get 6(x + 2y). This expression is simplified and easier to work with.

Common factors are also used in solving equations. When we are given an equation, we can factor out the common factor to simplify the equation and make it easier to solve. For example, consider the equation 6x + 9y = 15. Both terms have a common factor of 3, so we can divide both sides of the equation by 3 to get 2x + 3y = 5. This equation is simpler than the original equation and can be easily solved.

Another important concept related to common factors is prime factorization. Prime factorization is the process of breaking down a number into its prime factors, which are the factors that are prime numbers. For example, the prime factorization of 12 is 2 x 2 x 3, since 2 and 3 are prime factors of 12.

Prime factorization is useful in finding the GCF and the least common multiple (LCM) of two or more numbers. The LCM is the smallest number that is a multiple of two or more given numbers. To find the LCM, we need to find the prime factorization of each number and then multiply the highest powers of each prime factor. For example, consider finding the LCM of 12 and 18. The prime factorization of 12 is 2 x 2 x 3 and the prime factorization of 18 is 2 x 3 x 3. The highest power of 2 is 2, the highest power of 3 is 2, so the LCM of 12 and 18 is 2 x 2 x 3 x 3 = 36.

Definition:

A common factor is a number that is a factor of two or more numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Similarly, the factors of 18 are 1, 2, 3, 6, 9, and 18. Therefore, the common factors of 12 and 18 are 1, 2, 3, and 6.

Example 1:

Find the common factors of 24 and 36.

Solution:

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, while the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12.

Example 2:

Find the common factors of 40 and 60.

Solution:

The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40, while the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The common factors of 40 and 60 are 1, 2, 4, 5, 10, and 20.

Example 3:

Find the common factors of 12 and 30.

Solution:

The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The common factors of 12 and 30 are 1, 2, 3, and 6.

Example 4:

Find the common factors of 15 and 25.

Solution:

The factors of 15 are 1, 3, 5, and 15, while the factors of 25 are 1, 5, and 25. The common factors of 15 and 25 are 1 and 5.

Example 5:

Find the common factors of 8 and 16.

Solution:

The factors of 8 are 1, 2, 4, and 8, while the factors of 16 are 1, 2, 4, 8, and 16. The common factors of 8 and 16 are 1, 2, 4, and 8.