What are Factors and Multiples of a Number ?
Factors
- The numbers which exactly divides the given number are called the Factors of that number.
- Example :
Hence 1, 2, 3, 4, 6 and 12 are the factors of 12.
Multiples
If we say that 4 and 5 are the factors of 20 then 20 is the multiple of 4 and 5 both.- Multiples are always more than or equal to the given number.
Some facts about Factors and Multiples
- 1 is the only number which is the factor of every number.
- Every number is the factor of itself.
- All the factors of any number are the exact divisor of that number.
- All the factors are less than or equal to the given number.
- There are limited numbers of factors of any given number.
- All the multiples of any number are greater than or equal to the given number.
- There are unlimited multiples of any given numbers.
- Every number is a multiple of itself.
Perfect Number
- If the sum of all the factors of any number is equal to the double of that number then that number is called a Perfect Number.
Perfect Number Factors Sum of all the factors 6 1, 2, 3, 6 12 28 1, 2, 4, 7, 14, 28 56 496 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
Common Factors and Common Multiples
Example: 1What are the common factors of 25 and 55?
Solution:
Factors of 25 are 1, 5.
Factors of 55 are 1, 5, 11.
so Common factors of 25 and 55 are 1 and 5.
Example: 2
Find the common multiples of 3 and 4.
Solution:
Common multiples of 3 and 4 are 0, 12, 24 and so on.
Prime Factorisation
- Prime Factorisation is the process of finding all the prime factors of a number.
- There are two methods to find the prime factors of a number-
The prime factors of 36 are 2 and 3.
We can write 36 as a product of prime factors: 2 × 2 × 3 × 3
To
find prime factors using the repetitive division, it is advisable to
start with a small prime factor and continue the process with bigger
prime factors.
What is Highest Common Factor (HCF) ?
- The highest common factor (HCF) of two or more given numbers is the greatest of their common factors.
- Its other name is (GCD) Greatest Common Divisor.
Method to find HCF
Example
a) Find the HCF of 60 and 72.
Solution:
First, we have to find the prime factorisation of 60 and 72.
Then encircle the common factors.
b) Find the HCF of 18 & 21
| All the factors of 18 are 1, 2, 3, 6, 9, and 18 | All the factors of 21 are 1, 3, 7, and 21 |
|
1 ×18=18 2 × 9=18 3 × 6=18 |
1 × 21=21 7 × 3=21 |
It can be noticed that the highest common factor of 18 and 21 is 3.
Here, 3 is the largest number that divides all the given numbers. So, 3 is the HCF of 18 & 21.
Lowest Common Multiple (LCM)
- LCM stands for Lowest or Least Common Multiple.
- The lowest common multiple of two or more given number is the smallest of their common multiples.
- For example, find the LCM of 16 and 20. The LCM of 16 and 20 can be calculated as 2 × 2 × 2 × 2 × 5 = 80. Here, 80 is the LCM of numbers 16 and 20.
Methods to find LCM
1. Prime Factorisation Method
To find the LCM we have to find the prime factorisation of all the given numbers and then multiply all the prime factors which have occurred a maximum number of times.Example
Find the LCM of 60 and 72.
Solution:
First, we have to find the prime factorisation of 60 and 72.
Then encircle the common factors.
To find the LCM, we will count the common factors one time and multiply them with the other remaining factors.
LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 = 360
2. Repeated Division Method
If we have to find the LCM of so many numbers then we use this method.Example
Find the LCM of 105, 216 and 314.
Solution:
Use the repeated division method on all the numbers together and divide until we get 1 in the last row.
LCM of 105,216 and 314 is 2 × 2 × 2 × 3 × 3 × 3 × 5 × 7 × 157 = 1186920
Properties of HCF and LCM
a) The HCF of any two consecutive even numbers is always 2.
Explanation:
Let's assume the two consecutive even numbers are '2k' and '2k+2'
Write the numbers in factored form:
2k = 2 × k
2k+2 = 2 × (k+1)
The highest common factor of the numbers '2k' and '2k+2' is 2.
Example - Consider two consecutive even numbers, 4 and 6.
The HCF of 4 and 6 is 2.
b) HCF of any two consecutive numbers is equal to 1.
Consecutive numbers = 4, 5
Factors of 4 = 1,2,4
Factors of 5 = 1,5
Common Factor = 1
So, The Highest Common Factor (HCF) of 4 and 5 is 1
(c) The HCF of any two consecutive odd numbers is always 1.
Example: Consecutive odd numbers = 3, 5
Factors of 3 = 1 , 3
Factors 0f 5 = 1,5
Common Factors = 1
So, The HCF of 3 and 5 is 1.
d) The LCM of any two consecutive numbers is their product.
Consider two consecutive numbers say 10 and 11
The LCM of the above number is:
LCM (10, 11) = 10 * 11 = 110
e) The LCM of two consecutive even numbers is half their product
Example the LCM of 16 and 18 is equal to half into 16 into 18 is equal to 144
f) For two consecutive odd numbers the LCM is their product
Example LCM of 11 and 13 is 11 into 13 is equal to 143
g ) When one number is a factor of the other number then the smaller number is the HCF and the greater number is the LCM of the given numbers
Example for the numbers 12 and 24 ,12 is the HCF and 24 is the LCM
h) HCF = the product of common prime factors
i) LCM = product of common prime factors X the product of non common prime factors
Therefore LCM = HCF X the product of non-common prime factors
j)The product of LCM and HCF of any two given natural numbers is always equal to the product of those given numbers.
HCF X LCM = Number one X Number two
Some more properties of HCF and LCM
Property 1: The HCF of any given numbers is never greater than any of the numbers
The HCF of any given number is always less than or equal to any of the given numbers.
Example:
- Consider numbers 6 and 27. Their HCF is 3, which is a number that is lesser than both the numbers 6 and 27.
- Consider another example, say we have two numbers 4 and 20. Here the HCF is 4, which is equal to one of the numbers
- The HCF of 16, 18, and 24 is found to be 2. Here, 2 is less than all the given numbers
Property 2: The HCF of co-prime numbers is always 1
When both the numbers are co-prime, they have no common factor other than 1. Hence, 1 is the only common factor between the two given numbers. So, the HCF of two co-prime numbers is always 1.
Also, any two successive numbers/ integers are always co-prime. For instance, take any consecutive numbers such as 3, 4, or 2, 3, or 5, 6, and so on; they have 1 as their HCF. For instance, consider two co-prime numbers, 3 and 7. The HCF of 3 and 7 is 1.
Property 3: LCM of given co-prime numbers is always equal to the product of the numbers
Co-primes are the pair of numbers whose common factor includes just 1. Therefore, the LCM of two co-primes is always the product of these co-prime numbers. The LCM of a and b where a and b are co-primes is a × b.
For instance, Consider two co-prime numbers 3 and 5. The LCM of 3 and 5 is 3 × 5 × 1 = 15, which is the product of the co primes.
Property 4: H.C.F. and L.C.M. of Fractions
Let’s consider two fractions (p/q) and (r/s). To find LCM and HCF of (p/q) and (r/s) the generalized formula will be:
H.C.F = HCF of numerators / LCM of denominators
L.C.M = LCM of numerators / HCF of denominators
Example: Find the HCF and LCM of 4/9 and 6/21.
| Numerators of the two fractions are: 4 and 6 | Denominators of the two fractions are: 9 and 21 |
| Prime factors of 4 and 6: 4 = 2 × 2 6 = 2 × 3 |
Prime factors of 9 and 21: 9 = 3 × 3 21 = 3 × 7 |
| HCF of 4 and 6 is 2. LCM of 4 and 6 can be found as 2 × 2 × 3 = 12 |
HCF of 9 and 21 is 3. LCM of 9 and 21 can be found as 3 × 3 × 7 = 63. |
LCM of 4/9 and 6/21
Property 5: The LCM of given numbers is not less than any of the given numbers.
The LCM is the smallest number that both the given numbers divide into. However, it will be greater than at least one (or often both) of the given numbers.
Additionally, if a number is the factor of another number, then their LCM is the greater number itself.
For example, LCM of 8 and 12.
MUltiples of 8 = 8,16,24,32,40,48,56,64,72
Multiples of 12 = 12,24,36,48,60,72,84
Common Multiples = 24,48,72, ......
Least common multiple = 24
L.C.M. of 8 and 12 is 24 which is not less than any of the given numbers. For example, the LCM of 8 and 16 is 16, the greater number itself.
Important Notes
- H.C.F of two or more numbers is considered to be the greatest among all the common factors of them
- The LCM of two or more numbers can be defined as the smallest positive integer that is divisible by all the given numbers
- The HCF of co-prime numbers is always 1
- LCM of given co-prime numbers is always equal to the product of the number
Real life problems related to HCF and LCM
Example: 1There are two containers having 240 litres and 1024 litres of petrol respectively. Calculate the maximum capacity of a container which can measure the petrol of both the containers when used an exact number of times.
Solution:
As we have to find the capacity of the container which is the exact divisor of the capacities of both the containers, i. e. maximum capacity, so we need to calculate the HCF.
Example: 2
What could be the least number which when we divide by 20, 25 and 30 leaves a remainder of 6 in every case?
Solution:
As we have to find the least number so we will calculate the LCM first.
LCM of 20, 25 and 30 is 2 × 2 × 3 × 5 × 5 = 300.
Here
300 is the least number which when divided by 20, 25 and 30 then they
will leave remainder 0 in each case. But we have to find the least
number which leaves remainder 6 in all cases. Hence, the required number
is 6 more than 300.
The required least number = 300 + 6 = 306.
- L.C.M. of any two consecutive natural numbers is the same as their product because any common factor will divide their difference which is 1.
- H.C.F. of any two consecutive natural numbers is 1 because the difference between two consecutive numbers is 1.
Hence, L.C.M of two consecutive natural numbers is their product and H.C.F of two consecutive natural numbers is 1