To find the Highest Common Factor (HCF) of the numbers 910, 325, 615, 125, and 225, we will first find the prime factorization of each number.
### Step 1: Prime Factorization of Each Number
1. **For 910:**
- Divide by 2: \( 910 \div 2 = 455 \)
- Divide 455 by 5: \( 455 \div 5 = 91 \)
- Divide 91 by 7: \( 91 \div 7 = 13 \)
- Thus, the prime factorization of 910 is:
\[
910 = 2^1 \times 5^1 \times 7^1 \times 13^1
\]
2. **For 325:**
- Divide by 25: \( 325 \div 25 = 13 \)
- Thus, the prime factorization of 325 is:
\[
325 = 5^2 \times 13^1
\]
3. **For 615:**
- Divide by 5: \( 615 \div 5 = 123 \)
- Divide 123 by 3: \( 123 \div 3 = 41 \)
- Thus, the prime factorization of 615 is:
\[
615 = 5^1 \times 3^1 \times 41^1
\]
4. **For 125:**
- \( 125 = 5^3 \)
- Thus, the prime factorization of 125 is:
\[
125 = 5^3
\]
5. **For 225:**
- Divide by 25: \( 225 \div 25 = 9 \)
- Thus, the prime factorization of 225 is:
\[
225 = 5^2 \times 3^2
\]
### Step 2: Identify Common Factors
Now, we will look for the common prime factors in all the factorizations:
- **For 910:** \( 2^1 \times 5^1 \times 7^1 \times 13^1 \)
- **For 325:** \( 5^2 \times 13^1 \)
- **For 615:** \( 5^1 \times 3^1 \times 41^1 \)
- **For 125:** \( 5^3 \)
- **For 225:** \( 5^2 \times 3^2 \)
The only common prime factor across all numbers is \( 5 \).
### Step 3: Determine the HCF
The HCF is determined by taking the lowest power of the common prime factor:
- The lowest power of \( 5 \) in all the factorizations is \( 5^1 \).
Thus, the HCF of the numbers 910, 325, 615, 125, and 225 is:
\[
\text{HCF} = 5
\]
### Final Answer
The HCF of 910, 325, 615, 125, and 225 is **5**.
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