Find the greatest common factors of the following numbers. 1. 28 32 4) 12 16 20 2. 12 30 48 5) 8 56 80 112 3. 20 32 40
The Correct Answer and Explanation is:
Let’s go through the greatest common factor (GCF) calculations for each set of numbers.
1. GCF of 28, 32:
First, find the prime factorizations of both numbers:
- 28 = 2 × 2 × 7
- 32 = 2 × 2 × 2 × 2 × 2
The only common factor is 2 × 2, so the GCF is 4.
2. GCF of 12, 16, 20:
Prime factorizations:
- 12 = 2 × 2 × 3
- 16 = 2 × 2 × 2 × 2
- 20 = 2 × 2 × 5
The common factors are 2 × 2, so the GCF is 4.
3. GCF of 12, 30, 48:
Prime factorizations:
- 12 = 2 × 2 × 3
- 30 = 2 × 3 × 5
- 48 = 2 × 2 × 2 × 2 × 3
The common factors are 2 × 3, so the GCF is 6.
4. GCF of 8, 56, 80, 112:
Prime factorizations:
- 8 = 2 × 2 × 2
- 56 = 2 × 2 × 2 × 7
- 80 = 2 × 2 × 2 × 2 × 5
- 112 = 2 × 2 × 2 × 2 × 7
The common factors are 2 × 2 × 2, so the GCF is 8.
5. GCF of 20, 32, 40:
Prime factorizations:
- 20 = 2 × 2 × 5
- 32 = 2 × 2 × 2 × 2 × 2
- 40 = 2 × 2 × 2 × 5
The common factors are 2 × 2, so the GCF is 4.
Explanation:
To find the GCF, we start by finding the prime factorization of each number. This involves breaking each number down into its prime factors (smallest divisible numbers). Once we have the prime factorizations of all the numbers, we compare them to find the common factors.
For each number set, the common prime factors are identified and multiplied together to get the GCF. The highest number that divides all the numbers in the set without leaving a remainder is the greatest common factor.
