Factors and Multiples
Factors divide evenly into a number. Multiples are what you get when you multiply a number by whole numbers. The GCF (greatest common factor) and LCM (lowest common multiple) are the two most important applications , they are essential for working with fractions.
Factors of 12: 1, 2, 3, 4, 6, 12 (numbers that divide evenly into 12). Multiples of 12: 12, 24, 36, 48... (12 multiplied by 1, 2, 3, 4...). GCF(12, 8) = 4 , the largest number that divides both. LCM(12, 8) = 24 , the smallest number both divide into.
In this lesson
1 What Factors Are
A factor of a number is any whole number that divides into it evenly with no remainder. 4 is a factor of 20 because 20 ÷ 4 = 5 exactly. 6 is not a factor of 20 because 20 ÷ 6 = 3.33...
Factors always come in pairs: if 4 is a factor of 20, then 5 is too (because 4 × 5 = 20). To find all factors, find all such pairs.
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. A composite number has more than two factors. 1 is neither prime nor composite.
2 What Multiples Are
Multiples of a number are the results of multiplying it by 1, 2, 3, 4, and so on. Multiples of 6: 6, 12, 18, 24, 30, 36... Every number has infinitely many multiples.
A number is a multiple of another if it divides in evenly. 36 is a multiple of 6 (36 ÷ 6 = 6). 36 is also a multiple of 4 (36 ÷ 4 = 9). 36 is not a multiple of 8 (36 ÷ 8 = 4.5).
3 Finding the GCF (Greatest Common Factor)
The GCF of two numbers is the largest number that divides evenly into both. Also called the HCF (highest common factor).
4 Finding the LCM (Lowest Common Multiple)
The LCM of two numbers is the smallest number that both divide into evenly. Also called the LCD (lowest common denominator) when working with fractions.
For any two numbers a and b: GCF(a,b) × LCM(a,b) = a × b. So if you know one, you can find the other: LCM(24, 36) = (24 × 36) / GCF(24, 36) = 864 / 12 = 72.
5 Why GCF and LCM Matter
GCF is used to simplify fractions. To simplify 24/36, divide both by GCF(24,36) = 12: gives 2/3. You cannot simplify further because GCF(2,3) = 1.
LCM is used to add and subtract fractions. To add 1/4 + 1/6, find LCM(4,6) = 12. Convert: 3/12 + 2/12 = 5/12. The LCM becomes the common denominator.