## GCF of 28 and 42

The simplest way to determine the greatest common factor (GCF) of 28 and 42 is through listing their factors. The factors of 28 are 1, 2, 4, 7, 14, and 28, while the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. By comparing these lists, the common factors are 1, 2, 7, and 14. Among these, the largest is 14, making it the GCF of 28 and 42. This method of listing all factors is straightforward and effective, especially for smaller numbers, and quickly leads to identifying the highest number that both 28 and 42 can be divided by without a remainder.

## GCF of 28 and 42

### GCF of 28 and 42 is 14.

## GCF of 28 and 42 by Prime Factorization Method.

To find the greatest common factor (GCF) of 28 and 42 using the prime factorization method:

**Step 1:** Prime factorize both numbers:

**For 28:** 28 = 2² × 7

**For 42: **42 = 2 × 3 × 7

**Step 2:** Identify the common prime factors and their lowest powers:

The common prime factors between 28 and 42 are 2 and 7. The lowest power for each common factor is:

2¹ (since the lower of 2² and 2¹ is 2¹)

7¹ (since the lower of 7¹ in both factorizations is 7¹)

**Step 3: **Multiply the common prime factors with their lowest powers to determine the GCF:

**GCF** = 2¹ ×7¹ = 2 × 7 = 14

Therefore, the greatest common factor (GCF) of 28 and 42 by prime factorization method is 14.

## GCF of 28 and 42 by Long Division Method.

To find the greatest common factor (GCF) of 28 and 42 using the long division method:

**Step 1:** Start by dividing the larger number (42) by the smaller number (28).

42 ÷ 28 =1 with a remainder of 14.

**Step 2: **Then, take the divisors (28) and divide it by the remainder (14).

28 ÷ 14 = 2 with a remainder of 0.

Since the remainder is now 0, the division process stops here.

**Step 3: **The divisor at this step, where the remainder becomes zero, is the greatest common factor (GCF).

**GCF** = 14.

Therefore, the greatest common factor (GCF) of 28 and 42 by the long division method is 14.

## GCF of 28 and 42 by Listing Common Factors.

To find the greatest common factor (GCF) of 28 and 42 by listing common factors:

**Step 1:** List the factors of each number.

**Factors of 28**: 1, 2, 4, 7, 14, 28

**Factors of 42**: 1, 2, 3, 6, 7, 14, 21, 42

**Step 2: **Identify the common factors.

**Common factors**: 1, 2, 7, 14

**Step 3:** Determine the greatest common factor.

The highest number in the list of common factors is **14**.

## Can the Euclidean algorithm be used to find the GCF of 28 and 42?

Yes, repeatedly divide the larger number by the smaller number and use the remainders to find the GCF.

## Why is knowing the GCF of 28 and 42 important?

It helps in simplifying fractions, calculating ratios, and solving problems involving multiples.

## Can you use a calculator to find the GCF of 28 and 42?

Yes, many calculators have a GCF function, or you can use the prime factorization or Euclidean algorithm feature.

## What if the numbers were negative, would the GCF of -28 and -42 change?

No, the GCF is always considered as a positive number, regardless of the sign of the inputs.

## What is the relationship between the GCF and LCM of 28 and 42?

The product of the GCF and LCM of two numbers equals the product of the numbers.

## What advanced topics in mathematics require knowledge of the GCF?

Topics like simplifying algebraic expressions, solving diophantine equations, and understanding modular arithmetic require knowledge of the GCF.