To find the GCF (Greatest Common Factor) and LCM (Least Common Multiple) of the numbers 100, 140, and 250, we will follow these steps:
### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.
1. **Prime Factorization of 100:**
- 100 can be divided by 2: \(100 \div 2 = 50\)
- 50 can be divided by 2: \(50 \div 2 = 25\)
- 25 can be divided by 5: \(25 \div 5 = 5\)
- 5 can be divided by 5: \(5 \div 5 = 1\)
- Thus, the prime factorization of 100 is \(2^2 \times 5^2\).
2. **Prime Factorization of 140:**
- 140 can be divided by 2: \(140 \div 2 = 70\)
- 70 can be divided by 2: \(70 \div 2 = 35\)
- 35 can be divided by 5: \(35 \div 5 = 7\)
- 7 is a prime number: \(7 \div 7 = 1\)
- Thus, the prime factorization of 140 is \(2^2 \times 5^1 \times 7^1\).
3. **Prime Factorization of 250:**
- 250 can be divided by 2: \(250 \div 2 = 125\)
- 125 can be divided by 5: \(125 \div 5 = 25\)
- 25 can be divided by 5: \(25 \div 5 = 5\)
- 5 can be divided by 5: \(5 \div 5 = 1\)
- Thus, the prime factorization of 250 is \(2^1 \times 5^3\).
### Step 2: Finding the GCF
To find the GCF, we take the lowest power of all prime factors that appear in each factorization:
- For the prime factor 2:
- \(2^2\) (from 100), \(2^2\) (from 140), \(2^1\) (from 250) → the lowest power is \(2^1\).
- For the prime factor 5:
- \(5^2\) (from 100), \(5^1\) (from 140), \(5^3\) (from 250) → the lowest power is \(5^1\).
- For the prime factor 7:
- \(7^0\) (not present in 100), \(7^1\) (from 140), \(7^0\) (not present in 250) → the lowest power is \(7^0\).
Now, we multiply the lowest powers:
\[
\text{GCF} = 2^1 \times 5^1 = 2 \times 5 = 10.
\]
### Step 3: Finding the LCM
To find the LCM, we take the highest power of all prime factors that appear in any of the factorizations:
- For the prime factor 2:
- The highest power is \(2^2\) (from 100 and 140).
- For the prime factor 5:
- The highest power is \(5^3\) (from 250).
- For the prime factor 7:
- The highest power is \(7^1\) (from 140).
Now, we multiply the highest powers:
\[
\text{LCM} = 2^2 \times 5^3 \times 7^1 = 4 \times 125 \times 7.
\]
Calculating this step-by-step:
1. \(4 \times 125 = 500\)
2. \(500 \times 7 = 3500\)
### Final Answers
- **GCF of 100, 140, and 250 is 10.**
- **LCM of 100, 140, and 250 is 3500.**
