Lesson 4: The Highest Common Factor (H.C.F) and the Lowest Common Multiple (L.C.M)

1. Factors.
2. Prime and Composite Numbers.
3. Divisibility Rules.
4. Common Factors.
5. Highest Common Factor (H.C.F).
6. Finding H.C.F. of Big Numbers .
7. Multiples. Common Multiples.
8. Lowest Common Multiple (L.C.M.).
9. Finding L.C.M. of Big Numbers
10. Practice Game

 FACTOR

A factor of a given number is every number that divides exactly into that number.
 

Example

Write down all factors of 10.

    10 = 2 x 5, so numbers 2 and 5 are factors of 10.
    Also 10 = 10 x 1, so 10 and 1 are factors of 10.

   The factors of 10 are 1, 2, 5, 10.

NOTE: Number 1 and the number itself are always factors of any number.

PRIME AND COMPOSITE NUMBERS.

A prime number has exactly 2 factors, the number itself and 1.
In other words, the prime number can be divided only by 1 and by itself.
NOTE: 0 and 1 are not prime numbers.

Example: 5  is a prime number, because the only factors it has are 1 and 5.

The prime numbers less than 20 are 2,3,5,7,11,13,17,19
 

Example:

    Find all prime factors of 30.

    Solution:

    All the factors of 30 are 30, 15, 10, 6, 5, 3, 2, 1
    But only 5, 3 and 2 are prime numbers.
    Thefore all prime factors of 30 are 2, 3 and 5.

A composite number has at least one more factor  that the number itself or 1.
In fact, all whole numbers that are not prime are composite except for 1 and 0, which are not prime and not composite.

The composite numbers less than 20 are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
 

DIVISIBILITY RULES

The simple divisibility rules will help you to find factors of a number.

The number is divisible by:

• 2 if the last digit is 0, 2, 4, 6, or 8 (example: 12346);
• 3 if the sum of digits in the number are divisible by 3
  (example: 1236, because 1+2+3+6 = 12 = 3 x 4);
• 4 if the last 2 digits are divisible by 4
  (example: 897544, because 44 = 4 x 11);
• 5 if the last digit is 0 or 5
  (example: 178965 or 40980);
• 6 if it is divisible by 2 and 3;
• 7 sorry, no rule (you have to divide);
• 8 if the last 3 digits are divisible by 8
  (example: 124987080, because 080 = 8 x 10;
• 9 if the sum of digits is divisible by 9
  (example: 234612, because 2+3+4+6+1+2 = 18 = 9 x 2);
• 10 if the last digit is 0
  (example: 99990 );
• 100 if the last 2 digits are 0
  (example 987600);

NOTE: If a number is divisible by 2 factors, it is also divisible by the product of these  factors.

Example 1: Number 18 is divisible by 2 and 3, so it must be divisible by 2 x 3 = 6.

Example 2: Number 945 is divisible by 9 (why?) and by 5 (why?), so it must be divisible by 9 x 5 = 45. (Can you check it?)
 

COMMON FACTORS

When two (or more) numbers have the same factor, that factor is called a common factor.

Example

    Find all the common factors of 12 and 18.

    Factors of 12 are 1, 2, 3, 4, 6, 12.
    Factors of 18 are 1, 2, 3, 6, 18.
    The common factors of 12 and 18 are 1, 2, 3 and 6.
 

HIGHEST COMMON FACTOR (H.C.F).

The Highest Common Factor (H.C.F) of two (or more) numbers is the largest number that divides evenly into both numbers.
In other words the H.C.F is the largest of all the common factors.

The common factors or of 12 and 18 are 1, 2, 3 and 6.
The largest common factor is 6, so this is the H.C.F. of 12 and 18.

It is very easy to find a H.C.F. of small numbers, like 6 and 9 (it is 3) or 8 and 4 (it is 4).
The best way is to keep finding the factors of the smaller number, starting from the largest factor. The first factor of the smaller number that is also a factor of the larger number is a H.C.F.
 
The H.C.F is useful when simplifying fractions.

FINDING THE H.C.F. OF BIG NUMBERS

For larger numbers you can use the following method:

1. Find all prime factors of both numbers.
2. Write both numbers as a multiplication of prime numbers.
3. Find which factors are repeating in both numbers and multiply them to get  H.C.F (why?)

Example:

Find the Highest Common Factor (H.C.F.) of 240 and 924.

Solution:

Finding all prime factors of 240:

 We will start with the smallest prime number and we will divide 240 into it if we can (the divisibility rules come handy).
 We will do the same with the result (or quotient), and we will keep dividing by prime numbers until we have 1 as a quotient.
 Each time we write the prime factor to the right and the quotient below:
 

 
     240 = 2 x 2 x 2 x 2 x 3 x 5

NOTE: There are other methods for finding all prime factors of a number, for example a factor tree.
 
Finding all prime factors of 924:

          924 =  2 x 2 x 3 x 7 x 11

 Multiply the factors which repeat in both numbers  to get the H.C.F.

            The Highest Common Factor is 2 x 2 x 3 = 12
 

MULTIPLES. COMMON MULTIPLES.

When you multiply a given whole number by any other whole number, the result is a multiple of that number.
For example, 5 is the first multiple of 5 (because 5 x 1 = 5),
                    10 is the second multiple of 5, and so on.
Example 1:
    Write down the first 3 multiples of 8.
    Solution: 8 x 1 = 8,  8 x 2 = 16,  8 x 3 = 24, so the first 3 multiples of 8 are 8,16,24.

Example 2:
    Write down all multiples of 3 greater than 10 but smaller than 20.
    Solution: Multiples of 3 are 3, 6,9,12,15,18,21,24,...
                  The desired ones  are 12,15,18.

The common multiples of two numbers are multiples of both numbers.

Example 1:
    Find common multiples of 3 and 5.
    Solution: Multiples of 3 are 3,6,9,12,15,18,21,24,27,30,33,...
                  Multiples of 5 are 5,10,15,20,25,30,35,...
                 Common multiples of 3 and 5 are 15, 30, ...
 
 

LOWEST COMMON MULTIPLE (L.C.M.).

The Lowest Common Multiple (L.C.M) is the smallest number that is a common multiple of two or more numbers.

For example, the L.C.M of 3 and 5 is 15 .(see the example above).
 
The simple method of finding the L.C.M of smaller numbers is to write down the multiples of the larger number until one of them is also a multiple of the smaller number.
 

Example 1:

Find the Lowest Common Multiple of 8 and 12.

Solution: Multiples of 12 are 12, 24...
              24 is also a multiple of 8, so the L.C.M of 8 and 12 is 24.
 

FINDING L.C.M. OF BIG NUMBERS

1. Find all the prime factors of both numbers.
2. Multiply all the prime factors of the larger number by those prime factors of the smaller number that are not already included.

Example:

Find the Lowest Common Multiple (L.C.M.) of 240 and 924.

From the example of finding the H.C.F. we know the prime factors of both numbers.

924 = 2 x 2 x 3 x 7 x 11
240 = 2 x 2 x 2 x 2 x 3 x 5  

The L.C.M. is 2 x 2 x 3 x 7 x 11 x 2 x 2 x 5 = 924 x 20 = 18,480
 

Product

The result of multiplication. For example, the product of 4 and 5 is 20.