Show how to find all the factors of the following numbers in an efficient manner. Explain why you can stop checking for factors when you do. 63, 75, 126

To find all the factors of a given number efficiently, you don't need to check all the numbers up to that number. Instead, you only need to check up to the square root of the number. This is because factors come in pairs that multiply to give the original number, and one of each pair is less than or equal to the square root of the number.

### Steps to Find Factors Efficiently:

1. **Identify the Square Root:** Calculate the square root of the number.
2. **Check Divisibility:** Check each integer from 1 up to the square root of the number. If the number is divisible by an integer \(i\), then both \(i\) and \(\frac{\text{number}}{i}\) are factors.
3. **List All Factors:** If the quotient \( \frac{\text{number}}{i} \) is different from \(i\), both \(i\) and \( \frac{\text{number}}{i} \) are factors. If they are the same, include \(i\) only once.
4. **Stop After Square Root:** You can stop checking after reaching the square root because all larger factors would have already been identified as the pair of a smaller factor.

### Example 1: Factors of 63
1. **Square Root:** \( \sqrt{63} \approx 7.94 \), so check up to 7.
2. **Check Divisibility:**
   - \( 63 \div 1 = 63 \) → Factors: 1 and 63
   - \( 63 \div 2 \) (Not divisible)
   - \( 63 \div 3 = 21 \) → Factors: 3 and 21
   - \( 63 \div 4 \) (Not divisible)
   - \( 63 \div 5 \) (Not divisible)
   - \( 63 \div 6 \) (Not divisible)
   - \( 63 \div 7 = 9 \) → Factors: 7 and 9
3. **List All Factors:** 1, 3, 7, 9, 21, 63

### Example 2: Factors of 75
1. **Square Root:** \( \sqrt{75} \approx 8.66 \), so check up to 8.
2. **Check Divisibility:**
   - \( 75 \div 1 = 75 \) → Factors: 1 and 75
   - \( 75 \div 2 \) (Not divisible)
   - \( 75 \div 3 = 25 \) → Factors: 3 and 25
   - \( 75 \div 4 \) (Not divisible)
   - \( 75 \div 5 = 15 \) → Factors: 5 and 15
   - \( 75 \div 6 \) (Not divisible)
   - \( 75 \div 7 \) (Not divisible)
   - \( 75 \div 8 \) (Not divisible)
3. **List All Factors:** 1, 3, 5, 15, 25, 75

### Example 3: Factors of 126
1. **Square Root:** \( \sqrt{126} \approx 11.23 \), so check up to 11.
2. **Check Divisibility:**
   - \( 126 \div 1 = 126 \) → Factors: 1 and 126
   - \( 126 \div 2 = 63 \) → Factors: 2 and 63
   - \( 126 \div 3 = 42 \) → Factors: 3 and 42
   - \( 126 \div 4 \) (Not divisible)
   - \( 126 \div 5 \) (Not divisible)
   - \( 126 \div 6 = 21 \) → Factors: 6 and 21
   - \( 126 \div 7 = 18 \) → Factors: 7 and 18
   - \( 126 \div 8 \) (Not divisible)
   - \( 126 \div 9 = 14 \) → Factors: 9 and 14
   - \( 126 \div 10 \) (Not divisible)
   - \( 126 \div 11 \) (Not divisible)
3. **List All Factors:** 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126

### Why You Can Stop Checking at the Square Root:
- If \(n = a \times b\), and \(a \leq b\), then \(a \times a \leq n\), so \(a \leq \sqrt{n}\).
- Thus, once you've checked up to the square root, any larger factor \(b\) would have already been found as the pair of a smaller factor \(a\).

Would you like more details or have any other questions?

### 5 Related Questions:
1. How would you find the greatest common divisor (GCD) of 63, 75, and 126?
2. What is the least common multiple (LCM) of 63, 75, and 126?
3. How does the prime factorization of a number relate to its factors?
4. How can you use factorization to simplify fractions involving 63, 75, and 126?
5. Can you explain the concept of perfect squares and how they affect factorization?

### Tip:
Always start factorizing a number by checking the smallest primes (2, 3, 5, etc.), as it quickly narrows down the possible factors.

Learn how to efficiently find all factors of numbers 63, 75, and 126 using mathematical principles. Understand why you can stop checking for factors after reaching the square root of each number. Suitable for students in Grades 7-9.

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