Category : 5th Class
Factors and Multiples
Introduction
We have studied about the operations on numbers, in this chapter, we will study two important terms, that is, 'factors' and 'multiples'. They are related to the operations of multiplication and division.
Factors
Factors of a number is the number, which divides the given number completely. If a, b, c, d ?. are factors of 'm' then 'm' will be exactly divisible by a, b, c, d?.
How to Get Factors of a Number
To find all possible factors of a number, we have to find all the numbers, which divide the given number exactly.
Rules of Divisibility
(i) The numbers which have 0, 2, 4, 6, or 8 at the unit place is divisible by 2. Ex: 24545666, 69965654, 5484542130 are divisible by 2.
(ii) If sum of digits of a number is divisible by 3 then the number is divisible by 3. Ex: Sum of the digits of 25441215 = 2 + 5 + 4 + 4 + 1 + 2 + 1 = 24. 24is divisible by 3, therefore, 25441215 is divisible by 3.
(iii) If the number formed by its last two digits (ones and tens) is divisible by 4, the number is divisible by. Ex: 348568928 is divisible by 4 as 28 is the last two digits which are divisible by 4.
(iv) If a number has the digit 0 or 5 at unit's place, the number is divisible by 5. Ex: 5 is at the unit place in the number 544598265645, therefore, 544598265645 is divisible by 5.
(v) If a number is divisible by 2 as well as by 3, the number is divisible by 6. Ex: the number 6345822 is divisible by 6, since it is divisible by 2 as well as 3 as 2 is at unit?s place and sum of the number is 6 + 3 + 4 + 5 + 8 + 2 + 2 = 30, which is divisible by 3.
(vi) If the number formed by its last three digits is divisible by 8, the number is divisible by 8. Ex. 548602136 is divisible by 8. As the number formed by its last three digits is 136, which Is divisible by 8.
(vii) If sum of digits of a number is divisible by 9, the number is divisible by 9 Ex: sum of digits of 786545883 = 7 + 8 + 6 + 5 + 4 + 5 + 8 + 8 + 3 = 54 and 54 is divisible by 9. Thus 786545883 is divisible by 9.
(viii) If a number has the digit 0 at unit?s place, the number is divisible by 10 Ex: 0 is at the unit place in the number 2549896980, 2549896980 is divisible by 10:
Find all the possible factors of 15.
Solution: 1, 3, 5, 15 are factors of 15.
Find all the possible factors of 56.
Solution: 1, 2, 4, 7, 8, 14, 28, 56 are factors of 56.
Is 3 a factor of 4665366564?
Solution:
Yes.
Sum of digits of given number = 4 + 6 + 6 + 5 + 3 + 6 + 6 + 5 + 6 + 4 = 51 and 51 is divisible by 3.
Prime Number
The numbers which have only two factors, 1 and the number itself are called prime numbers.
For example:
Factors of 5 = 1, 5
Factors of 7 = 1, 7
Factors of 11 = 1, 11
Therefore, 5, 7, and 11 are prime numbers.
Twin Primes
Two consecutive prime numbers with the difference of 2 are called twin primes.
Write two pairs of twin primes.
Solution:
(3, 5), (5, 7) as 3, 5 and 7 are prime numbers and the two pairs of numbers have a difference of 2.
Composite Number
A number which has more than two factors is called a composite number,
Factors of 6 = 1, 2, 3, 6
Factors of 14 = 1, 2, 7, 14
Factors of 15 = 1, 3, 5, 15
Therefore, 6, 14 and 15 are composite numbers
Perfect Number
If sum of all the factors of a number is twice of the number, the number is called a perfect number
6 is a perfect number because sum of factors of 6 =1 + 2 + 3 + 6 = 12
Common Factors
The same factors of two or more than two different numbers are called common factors.
Factors of 12 = 1, 2, 3, 4, 6, 12,
Factors of 16 = 1, 2, 4, 8, 16
1, 2, 4 are the common factors of 12 and 16
Co-prime or Relatively Prime Numbers
If two numbers have only one common factor that is 1, the numbers are called co-prime or relatively prime numbers.
Factors of 8 = 1, 2, 4, 8
Factors of 9 = 1, 3, 9
1 is the only common factor of 8 and 9. Therefore, 8 and 9 are relatively co-prime numbers.
Factors of 18 = 1, 2, 3, 6, 9, 18
Factors of 27 = 1, 3, 9, 27
Common factors of 18 and 27 = 1, 3, 9
Multiples
When two or more than two numbers are multiplied with each other, the resulting number is the multiple of all that numbers. Like if \[A\times B=C,\] C is multiple of both A and B.
Multiples of 7 = 7, 14, 21, 28, 35, .......
Multiples of 8 = 8, 16, 24, 32, 40, ......
Common Multiples
The same multiples of two or more than two different numbers are called common multiples.
Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, .....
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, .....
Common multiples of 6 and 4 = 12, 24, 36 etc.
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ....
Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ....
Common multiples of 5 and 6 = 30, 60, .....
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