The Greatest Common Factor Of 12 And 42

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What is a Common Factor?

Before we dive into finding the greatest common factor of 12 and 42, it's important to understand what a common factor is. A factor is a number that divides another number evenly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. A common factor of two numbers is simply a factor that both numbers share.

How to Find Common Factors

To find the common factors of two numbers, you can list out all the factors of each number and look for the ones that they have in common. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. Looking at these lists, we can see that the common factors of 12 and 42 are 1, 2, 3, and 6.

What is the Greatest Common Factor?

The greatest common factor, also known as the greatest common divisor, is the largest factor that two numbers share. In this case, the greatest common factor of 12 and 42 is 6, since it is the largest factor that both numbers share.

Why is the Greatest Common Factor Important?

The greatest common factor is important in many areas of mathematics, including simplifying fractions and solving equations. When simplifying a fraction, you can divide both the numerator and denominator by their greatest common factor to reduce the fraction to its lowest terms. When solving equations, finding the greatest common factor can help you factor out common terms and simplify the problem.

How to Find the Greatest Common Factor

To find the greatest common factor of two numbers, you can use a few different methods. One common method is to list out all the factors of both numbers and circle the ones that they have in common. Then, you can choose the largest circled number as the greatest common factor. Another method is to use prime factorization, where you break down each number into its prime factors and look for the ones that they have in common.

Prime Factorization Method

To use the prime factorization method to find the greatest common factor of 12 and 42, we first need to break them down into their prime factors. The prime factors of 12 are 2, 2, and 3. The prime factors of 42 are 2, 3, and 7. To find the common factors, we can look for the prime factors that both numbers share. In this case, the only common prime factors are 2 and 3. To find the greatest common factor, we can multiply these common prime factors together, giving us 2 x 3 = 6.

Conclusion

In conclusion, the greatest common factor of 12 and 42 is 6. This can be found by listing out the common factors or by using prime factorization. The greatest common factor is important in many areas of mathematics and can be used to simplify fractions and solve equations.