FACTORS AND MULTIPLES

Intro

One of the branch of mathematics is most usually concerned with the numbers and various operations on numbers.There are two elementary concepts,known as “Factors and Multiples” studied in arithmetic.The proper knowledge of “factors and multiples ” is very important since these two concept are used in mathematics quite often.

Sample Problem

FACTORS
Factors are numbers divisible by another number exactly without any remainder.The factors of any number can be obtained in two ways which are :
(1.) By multiplication
(2.) By division
TYPES OF FACTORS
There are four types of factors namely ;
(1.) Prime numbers
(2.) Prime factors
(3.) Common factors
(4.) Highest common factors
PRIME NUMBERS
These are numbers divisible only by one and itself..A prime number must have two factors (i.e itself and 1 ).Examples of prime numbers are ;2 ,3 ,5 ,7 ,11 ,13 ,17 ,… 1 is not included because it has only one factor.
PRIME FACTORS
These are factors of the number that are prime. The prime factors any number can be obtained using “CONTINUED DIVISION METHOD”.This is done by dividing the giving number continuously starting with the lowest divisible prime number until a final 1 value is obtained.

COMMON FACTORS
These are numbers common to two or more giving numbers when expressed as a product of two or more numbers.The common factors of two or more numbers can be obtained in two ways which are ;
(1.) Prime factors method
(2.) Continued division method.

MULTIPLES
Multiples are what we get after multiplying by an integer (not a fraction).Multiples of any number can be obtained by multiplying the number by 1, 2 , 3 , 4 , 5 ,…( and so on ).First ten multiples of 2 are ; 2 , 4 , 6 ,8 ,10 ,12 , 14 , 16 , 18 and 20.
Similarly , multiples of 5 are ; 5 ,10 , 15 ,20 ,25 , 30 ,35 , 40 ,45 ,50 ,55 ,60 and so on .
TYPES OF MULTIPLES
There are two types of multiples namely :
(1.) COMMON MULTIPLES
(2.) LOWEST COMMON MULTIPLES

COMMON MULTIPLES
To find the common multiples of a set of numbers , multiply each of the set of numbers by 1 , 2 , 3 , 4 , … etc and check the common multiples.

LOWEST COMMON MULTIPLES ( L.C.M )
To find the L.C.M of a given set of numbers , divide each number by some set of prime numbers repeatedly until the result of the division is one ,then the product of each set of prime numbers gives the L.C.M

Solution

Example 1. Find the factors of 24 using the two methods .
Using multiplication method Using division method
1 x 24 = 24 24 ÷ 1 =24
2 x 12 = 24 24 ÷ 2 =12
3 x 8 = 24 24 ÷ 3 =8
4 x 6 = 24 24 ÷ 6 =4
Factors of 24 are :1 ,2 ,3 ,4 ,6 ,8 ,12 , and 24. Factors of 24 are ;1 ,2 ,3 ,4 ,6 ,8 ,12 ,and 24.
Note:For each method ,the answer is obtained starting from 1 to the highest value 24 and repetition must be avoided.

EXAMPLE 2.Express 72 as a product of prime factors.
Using continued division method
2 | 72
2 | 36
2 | 18
3 | 9
3 | 3
1
The prime factors of 72 are 2 x 2 x 2 x 3 x 3.

EXAMPLE 3;Find the common factors 36 ,40 ,and 72
Using prime factor method using continued division method
36 =2 x 2 x 3 x 3 2 | 36 | 40 | 72 |
40 =2 x 2 x 2 x 5 2 | 18 | 20 | 36 |
72= 2 x 2 x 2 x 3 x 3 | 9 | 10 | 18 |

Common factors of 36 ,40 , and 72 are 2 x 2.
Note 1:when using the prime factor method ,each of the given numbers must be expressed as product of prime factors. then the common factor is 2 x 2 .

Note 2:when using the continued division method ,a common prime number(lowest prime numbers must always be considered first) should be used to divide the numbers until there is no common factor again.common factor is then 2 x 2 .

HIGHEST COMMON FACTORS
This is the product of the common factors of any two or more giving numbers.The highest common factors of 36 ,40 ,and 72 is therefore the product of the common factors ; 2 x 2 which is 4 .

EXAMPLE 4 : Write out the first three common multiples of 3 and 4.

Multiples of 3 : 3 , 6 , 9 ,(12), 15 , 18 , 21 ,(24), 27 , 30 , 33 ,(36).
Multiples of 4 : 4 , 8 ,(12), 16 , 20 ,(24), 28 , 32 ,(36), 40 , 44 , 48.
First three common multiples Of 3 and 4 are : 12 , 24 , 36 .

EXAMPLE 5 : Find the L.C.M of the following (a.) 8 , 9 , and 12 (b.) 24 , 28 , and 50.
(a.) (b. )
2 | 8 | 9 | 12 | 2| 24 | 28 | 50 |
2 | 4 | 9 | 6 | 2| 12 | 14 | 25 |
2 | 2 | 9 | 3 | 2| 6 | 7 | 25 |
3 | 1 | 9 | 3 | 3| 3 | 7 | 25 |
3 | 1 | 3 | 1 | 5| 1 | 7 | 25 |
| 1 | 1 | 1 | 5| 1 | 7 | 5 |
The LCM is 2 x 2 x 2 x 3 x 3 = 72 7| 1 | 7 | 1 |
| 1 | 1 | 1 |
The LCM is 2 x 2 x 2 x 3 x 5 x 5 x 7 = 4200



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