Understanding Factors

A factor (or divisor) of a number is an integer that divides that number exactly, with no remainder. Understanding factors is fundamental to many areas of mathematics.

Finding Factors

To find all factors of a number n:

1. Start with 1 (always a factor)
2. Check each number up to √n to see if it divides n evenly
3. For each factor found, n divided by that factor is also a factor

Factor Pairs

Factor pairs are two numbers that multiply together to give the original number. For example, factor pairs of 12 are:

• 1 × 12 = 12
• 2 × 6 = 12
• 3 × 4 = 12

Prime Factorization

Every positive integer greater than 1 can be expressed as a product of prime numbers in exactly one way (Fundamental Theorem of Arithmetic).

• 12 = 2² × 3
• 100 = 2² × 5²
• 360 = 2³ × 3² × 5

GCD and LCM

GCD (Greatest Common Divisor): The largest number that divides both numbers evenly. Also called HCF (Highest Common Factor).

LCM (Least Common Multiple): The smallest number that is a multiple of both numbers.

For any two numbers a and b: GCD(a, b) × LCM(a, b) = a × b

Number Classifications

• Perfect Number: Sum of proper divisors equals the number (e.g., 6 = 1 + 2 + 3)
• Deficient Number: Sum of proper divisors is less than the number
• Abundant Number: Sum of proper divisors is greater than the number

Perfect Numbers

The first few perfect numbers are:

• 6 = 1 + 2 + 3
• 28 = 1 + 2 + 4 + 7 + 14
• 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
• 8,128