what is the greatest common factor of 540 and 600
what is the greatest common factor of 540 and 600
The Correct Answer and Explanation is:
The greatest common factor (GCF) of 540 and 600 is 60.
Explanation
To understand why 60 is the greatest common factor (GCF) of 540 and 600, we must first understand what a GCF is. The GCF of two numbers is the largest number that divides both numbers without leaving a remainder.
There are a few ways to find the GCF, but the most common is by prime factorization.
Step 1: Prime factorization
540
First, divide by 2 (the smallest prime number):
- 540 ÷ 2 = 270
- 270 ÷ 2 = 135
Next, divide 135 by 3: - 135 ÷ 3 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
Now 5 is a prime number.
So,
540 = 2² × 3³ × 5
600
- 600 ÷ 2 = 300
- 300 ÷ 2 = 150
- 150 ÷ 2 = 75
Now divide by 3: - 75 ÷ 3 = 25
Then divide 25 by 5: - 25 ÷ 5 = 5
- 5 ÷ 5 = 1
So,
600 = 2³ × 3 × 5²
Step 2: Identify common prime factors
Now compare the prime factorizations:
- 540 = 2² × 3³ × 5
- 600 = 2³ × 3 × 5²
Take the lowest powers of each common prime factor:
- 2: min(2, 3) = 2 → 2²
- 3: min(3, 1) = 1 → 3¹
- 5: min(1, 2) = 1 → 5¹
Multiply these together:
GCF = 2² × 3 × 5 = 4 × 3 × 5 = 60
Final Answer:
✅ The greatest common factor of 540 and 600 is 60.
This method ensures we accurately determine the highest number that divides both integers completely
.