What is the Greatest Common Factor (GCF) of 92 and 120?

Let's explore the greatest common factor (GCF) of two numbers, particularly 92 and 120. In simpler terms, the GCF is the largest number that can divide both 92 and 120 without leaving any remainder.

Understanding GCF and Common Factors

The GCF, also known as the highest common factor (HCF), represents the largest number that is a factor of both 92 and 120. To find this, we can follow a few steps:

Identify the factors of each number. Determine the common factors. Select the highest of these common factors.

Prime Factorization Method

For the prime factorization method, we start by breaking down each number into its prime factors.

Prime Factors of 92

92 is divisible by 2, so we write it as 92 2 × 46. 46 is also divisible by 2, so we further break it down to 46 2 × 23. 23 is a prime number, so 23 cannot be factored further.

Thus, the prime factorization of 92 is: 92 22 × 23.

Prime Factors of 120

120 is divisible by 2, so we write it as 120 2 × 60. 60 is also divisible by 2, so we write it as 60 2 × 30. 30 is also divisible by 2, so we write it as 30 2 × 15. 15 is divisible by 3, so we write it as 15 3 × 5. 3 and 5 are both prime numbers.

Thus, the prime factorization of 120 is: 120 23 × 3 × 5.

Common Prime Factors

From the prime factorization, we see that the common prime factors of 92 and 120 are 22. This means that the GCF of 92 and 120 is 22, which equals 4.

Therefore, the GCF of 92 and 120 is 4. This can be expressed as:

GCF(92, 120) 4

Verification

We can verify this by finding the factors of each number and checking for the highest common factor:

Factors of 92

1, 2, 4, 23, 46, 92

Factors of 120

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

The common factors of 92 and 120 are 1, 2, 4. Among these, the highest common factor is 4.

Conclusion

In conclusion, the greatest common factor (GCF) of 92 and 120 is 4, which is the largest number that can divide both 92 and 120 without leaving a remainder. This is a fundamental concept in mathematics, especially in simplifying fractions and solving equations involving integers.

By understanding and applying the prime factorization method, we can easily find the GCF of any two numbers, making it a valuable tool in various mathematical contexts.

Keywords: greatest common factor, finding GCF, prime factorization