LCM AND GCF

Factor

When a whole number is divided into another whole number without a remainder, it is called a factor.

Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite).

Example: The number 6 is a factor 54, because it goes into 54 evenly 9 times. In this case 9 is also a factor of 54.

Example:2 and 3 are factors of 6,because 2 × 3 = 6.

Example:3 is a factors of 9, because 3 × 3 = 9. 

Note: Prime numbers only have 2 factors: 1 and itself

Prime Factors

All factors of a number that are prime are called prime factors.

Prime Factorization

The Prime Factorization of a number is written as the product of all its prime factors.

Common Factor

A common factor is a whole number that is a factor of two or more nonzero whole numbers.

Example: 3 is a common factor of the numbers 15 and 24. Both 15 and 24 have 3 as a factor, therefore 3 is called “common” between 15 and 24.

  

Greatest Common Factor

The greatest common factor, or GCF, is the greatest factor that divides two numbers.

Different methods to find GCF

There are 3 methods for finding the GCF.

1   Method 1. Rainbow method 

Example: Find the GCF of 14, 35.

List all the factors

14: 1, 2, 7, 14

35: 1, 5, 7, 35

Common factors: 1, 7

GCF: 7

     Method 2.  Factor Tree method 

                Example: Find the GCF of 18, 24

                step 1: List all the factors using factor tree

                           18 = 2x3x3,  24 = 2x2x2x3

                step 2: Identify the common factors

                           The common factors of 18 and 24 are 2 and 3. 

                Step 3: Multiply the common factors to find the GCF.

                           GCF = 3x2 = 6

            Method 3. CAKE or ladder Method

      

Step 1:

Divide the numbers by the prime number until the numbers cannot be divided evenly

Step 2:

     Multiply all prime numbers used to divide. 

Multiple

The multiple, x, of a number, n, is a number where n is a factor of x.

Example: The number 33 (x) can be divided by 3 (n) without any remainder. So 33 is a multiple of 3 (3 * 11 = 33).

Common multiple

A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 12, 24, 36...

Least common multiple (LCM)

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers.

Example: The LCM of 2, 3, 4, and 6 is 12.

Different methods to find LCM

Method 1: Listing  Multiples

Find LCM of 3 and 4

Step 1:

List all multiples of 3 and 4

Step 2: 

find common multiples

Step 3:

Find least common multiple

Method 2: Using Prime Factorization

Step 1: 

Identify Common factors 

common factors for 18 and 24 are 3 and 2

step 2:

Multiply common factors with all other factors

Other factors for 18 is 3 , other factors for 24 are 2 and  2

Step 3:

Multiply the common factors with other factors

LCM = 2x3x3x2x2= 72

Method 3: ladder method

Relationship between the GCF and LCM

The product of the GCF and the LCM of two or more numbers is equal to the product of the numbers:

      GCF (a, b) · LCM (a, b) = a · b

      LCM (a,b) is read as "LCM of numbers a and b"

      GCF (a,b) is read as "GCF of numbers a and b"

Example:

    GCF (12, 16) = 4

   LCM (12, 16) = 48

   48 · 4 = 12 ·16

   192 = 192