Arithmetic: Finding Factors of a Number, Prime Numbers, Composite Numbers

A factor of a number is a whole number that divides evenly into that number without leaving a remainder. In other words, if A × B = C, then both A and B are factors of C.

How to Find Factors of a Number

Step 1: Start with 1 and the Number Itself

• Every number has at least two factors: 1 and itself.
• Example: Factors of 12 must include 1 and 12.

Step 2: Check for Divisibility

• Divide the number by 2, 3, 4, etc. to see if it results in a whole number.
• If yes, both the divisor and quotient are factors.

Step 3: Stop at the Square Root

• You only need to check numbers up to the square root of the given number.
• Example: For 36, check divisibility up to 6 because - Square root 36=6.

Example 1: Finding Factors of 18

1. Start with 1 and 18 → (1, 18)
2. Check divisibility:
   • 18 ÷ 2 = 9 → (2, 9)
   • 18 ÷ 3 = 6 → (3, 6)
3. Stop at the Square root 

✅ Factors of 18: {1, 2, 3, 6, 9, 18}

Example 2: Finding Factors of 36

1. Start with 1 and 36 → (1, 36)
2. Check divisibility:
   • 36 ÷ 2 = 18 → (2, 18)
   • 36 ÷ 3 = 12 → (3, 12)
   • 36 ÷ 4 = 9 → (4, 9)
   • 36 ÷ 6 = 6 → (6, 6)
3. Stop at 6 (since Sq root )

✅ Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}

Prime Numbers vs. Composite Numbers

• Prime Numbers: Only have two factors (1 and itself).
  • Example: 7 → Factors: {1, 7}
• Composite Numbers: Have more than two factors.
  • Example: 12 → Factors: {1, 2, 3, 4, 6, 12}

Special Case: Finding Factors of Large Numbers

For big numbers, use prime factorization or division method to systematically find factors.

Example: Finding factors of 100

• Prime factorization: $100 = 2^2 \times 5^2$
• Possible factor pairs: (1,100), (2,50), (4,25), (5,20), (10,10)

✅ Factors of 100: {1, 2, 4, 5, 10, 20, 25, 50, 100}

Practice Problems

1. Find all factors of 24.
2. List the factors of 45.
3. What are the factors of 50?
4. Identify all factors of 81.
5. Find the factors of 120.