what is the gcf of 46 and 60

Natashya

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02-02-2025

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The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest number that divides evenly into two or more numbers. Let's find the GCF of 46 and 60. We'll explore a couple of methods to achieve this.

Method 1: Listing Factors

This method involves listing all the factors of each number and then identifying the largest factor they share.

Factors of 46: 1, 2, 23, 46

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Comparing the two lists, we see that the largest number present in both lists is 2.

Therefore, the GCF of 46 and 60 is $\boxed{2}$.

Method 2: Prime Factorization

This method is particularly useful for larger numbers. It involves breaking down each number into its prime factors. Prime factors are numbers only divisible by 1 and themselves (e.g., 2, 3, 5, 7, etc.).

Prime Factorization of 46:

46 = 2 x 23

Prime Factorization of 60:

60 = 2 x 2 x 3 x 5 = 2² x 3 x 5

Now, we identify the common prime factors and their lowest powers. Both 46 and 60 share a single factor of 2 (2¹).

Therefore, the GCF of 46 and 60 is 2¹ = $\boxed{2}$.

Method 3: Euclidean Algorithm (For Larger Numbers)

The Euclidean algorithm is a more efficient method for finding the GCF of larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCF.

1. Divide the larger number (60) by the smaller number (46): 60 ÷ 46 = 1 with a remainder of 14

2. Replace the larger number with the smaller number (46) and the smaller number with the remainder (14): 46 ÷ 14 = 3 with a remainder of 4

3. Repeat the process: 14 ÷ 4 = 3 with a remainder of 2

4. Repeat again: 4 ÷ 2 = 2 with a remainder of 0

The last non-zero remainder is 2. Therefore, the GCF of 46 and 60 is $\boxed{2}$.

Conclusion

Regardless of the method used, the greatest common factor of 46 and 60 is 2. Choosing the best method depends on the size of the numbers involved and your personal preference. For smaller numbers, listing factors is often the quickest approach. For larger numbers, prime factorization or the Euclidean algorithm are more efficient.