To find the common factors of the numbers 60, 75, and 105, we will follow these steps:
### Step 1: Find the factors of 60
- The factors of 60 are the numbers that divide 60 without leaving a remainder.
- Start with 1: \(60 \div 1 = 60\)
- Next, check 2: \(60 \div 2 = 30\)
- Check 3: \(60 \div 3 = 20\)
- Check 4: \(60 \div 4 = 15\)
- Check 5: \(60 \div 5 = 12\)
- Check 6: \(60 \div 6 = 10\)
- Check 10: \(60 \div 10 = 6\)
- Check 12: \(60 \div 12 = 5\)
- Check 15: \(60 \div 15 = 4\)
- Check 20: \(60 \div 20 = 3\)
- Check 30: \(60 \div 30 = 2\)
- Check 60: \(60 \div 60 = 1\)
**Factors of 60:** 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
### Step 2: Find the factors of 75
- The factors of 75 are the numbers that divide 75 without leaving a remainder.
- Start with 1: \(75 \div 1 = 75\)
- Check 3: \(75 \div 3 = 25\)
- Check 5: \(75 \div 5 = 15\)
- Check 15: \(75 \div 15 = 5\)
- Check 25: \(75 \div 25 = 3\)
- Check 75: \(75 \div 75 = 1\)
**Factors of 75:** 1, 3, 5, 15, 25, 75
### Step 3: Find the factors of 105
- The factors of 105 are the numbers that divide 105 without leaving a remainder.
- Start with 1: \(105 \div 1 = 105\)
- Check 3: \(105 \div 3 = 35\)
- Check 5: \(105 \div 5 = 21\)
- Check 7: \(105 \div 7 = 15\)
- Check 15: \(105 \div 15 = 7\)
- Check 21: \(105 \div 21 = 5\)
- Check 35: \(105 \div 35 = 3\)
- Check 105: \(105 \div 105 = 1\)
**Factors of 105:** 1, 3, 5, 7, 15, 21, 35, 105
### Step 4: Identify the common factors
- Now we will compare the factors of 60, 75, and 105 to find the common ones.
- **Factors of 60:** 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- **Factors of 75:** 1, 3, 5, 15, 25, 75
- **Factors of 105:** 1, 3, 5, 7, 15, 21, 35, 105
The common factors are those that appear in all three lists:
- Common factors: 1, 3, 5, 15
### Final Answer:
**The common factors of 60, 75, and 105 are:** 1, 3, 5, and 15.
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