Category : 4th Class
Factors and Multiples
Factors of a Number
All the numbers, which divide a certain number exactly, without leaving a remainder are called factors of that number.
For example:
\[\Rightarrow \]1, 2, 3, 4, 6 and 12 are factors of 12.
Note: Factors of a number always include 1 and the number itself.
Find the factors of 15.
Solution: The factor of 15 are
\[\Rightarrow \]1, 3, 5 and 15 are factors of 15.
Which among the following is not a factor of 10?
(a) 2 (b) 5
(c) 10 (d) 3
(e) None of these
Answer (d)
Explanation:
Clearly \[10\div 1=10,\,10\div 2=5,\,10\div 5=2\,and\,10\div 10=1\]\[\Rightarrow \]1, 2, 5 and 10 are factors of 10.
Properties of factors:
(i) 1 is a factor of every number.
(ii) Every non-zero number is a factor of intself.
(iii) Every non-zero number is a factor of zero.
(iv) Division by 0 is meaningless.
(v) The factor of a non-zero number is either less than or equal to the number.
Which among the following statements is not ture?
(a) 2 is a factor of 2.
(b) 26 is a factor of 0
(c) 28 is not a factor of 4.
(d) 4 is not a factor of
(e) None of these
Answer (d)
Explanation: Every number is a factor of itself so 2 is a factor of 2.
Every non-zero number is a factor of 0. So 26 is a factor of 0.
The factor of a non-zero number cannot be greater than the number.
So, 28 can't be a factor of 4.
\[28=1\times 2\times 2\times 7\]
\[\Rightarrow \]1, 2, 4, 7, 14 and 28 are factors of 28.
Even and odd Numbers
Even numbers: A number is called an even number if 2 is a factor of the number. In other words, A number, which is a multiple of 2 is called an even number.
For example: 0, 2, 4, 6, 8, 10, 12, 14, 16 are even numbers.
Odd numbers: A number, which is not a multiple of 2 is called an odd number.
For example 1, 3, 5, 7, 9, 11, —— are odd numbers.
Which one among the following is not an even number?
(a) 0 (b) 89990
(c) 1049 (d) 2032
(e) None of these
Answer (c)
Explanation: 1049 is not a multiple of 2 and so is not an even number.
Prime Factors
Factors of a number written in primes are called prime factors of that number.
For example: \[24=2\times 2\times 2\times 3\]
\[\Rightarrow \]Prime factors of 24 are \[2\times 2\times 2\times 3\]
Multiples
You already know that multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ——. So all the numbers that comes in the table of 2 are its multiples. Multiples are never ending.
So it is not possible to find the last multiple of any number.
Write first five multiples of 8 and 16 and find common multiples of both.
Solution: First five multiples of 8 = 8, 16, 24, 32, 40
First five multiples of 16 = 16, 32, 48, 64, 80
Clearly common multiples from first five multiples of 8 and 16 are 16 and 32.
Which one among the following is not a multiple of II?
(a) 101 (b) 121
(c) 154 (d) 176
(e) None of these
Answer (a)
Explanation:\[121=11\times 11,154=11\times 14\]\[176=11\times 16\,and\,101=1\times 101\]
Here 101 is a prime number and it is not a multiple of 11.
Properties of Multiples
(i) Every number is a multiple of 1.
(ii) Every non-zero number is a multiple of itself.
(iii) Multiples of any number are infinite.
(iv) Every non-zero multiple of a non-zero number is either greater than or equal to the number.
Highest Common Factor
The highest common factor among two or more given numbers is called the highest common factor or (H.C.F.).
Find the HCF of 12 and 16.
Solution:\[\because 12=2\times 2\times 3\,and\,16=2\times 2\times 2\times 2\]
\[\because \]HCF of 12 and \[16=2\times 2=4\]
AlterNet Method
Factors of 12=1, 2, 4, 6, 12 \[\{\because 12=1\times 2\times 2\times 3\}\] and
Factors of 16=1, 2, 4, 8, 16 \[\{\because 16=1\times 2\times 2\times 2\times 2\}\]
\[\therefore \]Highest common factors of 12 and 16 = 4
Lowest Common Multiple (L.C.M.)
Since multiples of a number are uncountable. So it is not possible to get the highest common multiple. Let us learn the steps used to find the lowest common multiple of two numbers.
Step 1: Find first few multiples of smaller number.
Step 2: Find first few multiples of larger number till we get a common multiple of both the numbers.
Step 3: The common multiple so obtained will be the lowest common multiple of both the numbers.
Find the lowest common multiple of 4 and 10.
Solution:
Step 1: First 6 multiples of 4 are 4, 8, 12, 16, 20,
24
Step 2: First 2 multiples of 10 are 10, 20
Step 3: Clearly 20 is the first common multiple of both the numbers.
So, 20 is the LCM of 4 and 10.
Alternate Method
First write both the numbers separated with a comma, and then find the prime factors of both the number as shown below.
|
2 |
4, 10 |
|
2 |
2, 5 |
|
5 |
1, 5 |
|
|
1, 1 |
So LCM of 4 and 10 is \[2\times 2\times 5=20\]
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