what is the gcf of 125 100 and 125 with a solution
The Correct Answer and Explanation is:
Correct Answer:
The greatest common factor (GCF) of 125, 100, and 125 is 25.
Solution:
We are asked to find the greatest common factor (GCF) of 125, 100, and 125.
Step 1: Find the prime factorization of each number
- 125
125 is divisible by 5:
125 ÷ 5 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
So, 125 = 5 × 5 × 5 = 5³ - 100
100 ÷ 2 = 50
50 ÷ 2 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
So, 100 = 2² × 5²
Step 2: Identify common prime factors
We compare the prime factors of:
- 125 = 5³
- 100 = 2² × 5²
The only common prime factor is 5, and we take the lowest power of this factor among the numbers.
- 5 appears as 5³ in 125 and 5² in 100
- The smallest power is 5² = 25
So, GCF = 25
We can ignore the repeated 125, since the GCF of 125, 100, and 125 is the same as the GCF of just 125 and 100.
Explanation
The greatest common factor (GCF) of a set of numbers is the largest number that can divide all the given numbers exactly, without leaving a remainder. This is especially useful when simplifying fractions, factoring polynomials, or solving word problems involving groups.
To find the GCF of 125, 100, and 125, we start by performing a prime factorization of each number. Prime factorization breaks down a number into its building blocks—its prime numbers. For 125, the prime factor is 5. Dividing repeatedly by 5 gives 125 = 5 × 5 × 5 or 5³. For 100, dividing by its smallest prime factors, we get 100 = 2 × 2 × 5 × 5 or 2² × 5².
Next, we identify the common prime factors shared by all numbers. Both 125 and 100 share the prime factor 5. While 125 has 5³ and 100 has 5², we select the lowest power of the common factor, which is 5² = 25. This is because a higher power would not divide evenly into the number with the smaller power.
The number 25 can divide both 125 and 100 exactly, and since it is the largest such number, it is the GCF.
Therefore, the greatest common factor of 125, 100, and 125 is 25.
