1. Relationship between HCF and LCM

2. Recap-Highest Common Factor
Definition: The Highest Common Factor (HCF) of two or more given numbers is the highest (or greatest) of their common factors. It is also known as Greatest Common Divisor (GCD). Find the Highest Common Factor(HCF) of 12 and 16. Factors of 12 : 1, 2, 3, 4, 6, 12 Factors of 16: 1, 2, 4, 8, 16 The common factors of 12 and 16 are 1 , 2 and 4 Highest Common Factor(HCF) will be the highest of the given common factors – i.e. 4 Therefore HCF of 12 and 16 is 4.

3. Recap-Lowest Common Multiple
Definition: The Lowest Common Multiple (LCM) of two or more given numbers
is the lowest (or smallest or least) of their common multiples.
Example: What is the Lowest Common Multiple of 2 and 3?
Multiples of 2: 2 , 4 , 6 , 8 , 10 ,12 ,…
Multiples of 3: 3 , 6 , 9 , 12 ,15 ,….
The common multiples of 2 and 3 are 6, 12 etc…..
The Lowest Common Multiple will be the lowest of their common multiples .i.e. 6.
Therefore LCM of 2 and 3 is 6.

4. Relationship between HCF and LCM
The product of highest common factor (H.C.F.) and lowest common multiple
(L.C.M.) of two numbers is equal to the product of two numbers.
H.C.F × L.C.M = product of two numbers
For example : Take two numbers 15 and 18
HCF of 15 and 18 is 3 , LCM of 15 and 18 is 90
H.C.F. × L.C.M. = Product of the 2 given numbers
3 × × 18
270 = 270
Therefore , product of H.C.F. and L.C.M. of 15 and 18 = product of 15 and 18.

5. In order to solve problems consisting of product of numbers and
HCF, LCM let us recall the relationship between Multiplication
and Division
Multiplication
3 x 4 = 12
Division fact
÷ 4 = 3
12 ÷ 3 = 4
Division
12 ÷ 3 = 4
Multiplication fact
3 X 4 = 12
4 X = 12

6. Example 1: The LCM of 24 and 45 is 360. Find the HCF? Solution:
Given:- Numbers =24 and 45 , LCM = 360
To Find:- HCF
We know HCF x LCM = product of two numbers
? x = ( x )
? x =
Therefore ? = ÷ 360
So, HCF (24 , 45) = 3
Ans: 3
We know if 3 x 4 = 12 then :
12 ÷ 4 = 3
12 ÷ 3 = 4

7. Example 2: The HCF of 12 and 18 is 6. Find the LCM? Solution:
Given:- Numbers =12 and 18 , HCF = 6
To Find:- LCM
We know HCF x LCM = product of two numbers
6 x ? = ( x )
6 x ? =
Therefore ? = ÷ 6
So, LCM (12 , 18) = 36
Ans: 36
We know if 3 x 4 = 12 then :
12 ÷ 4 = 3
12 ÷ 3 = 4

8. Given:- LCM=300 , HCF=50 , Known Number = 150 To Find:- Unknown number
Example 3: The L. C. M. and H. C. F. of two numbers are 300 and 50 respectively. If one of the number is 150 find the other number?
Solution:
Given:- LCM=300 , HCF=50 , Known Number = 150
To Find:- Unknown number
We know HCF x LCM = product of two numbers
300 x = ( 150 x ? )
= x ?
15000 ÷150 = ?
So, Unknown number (?) = 100
Ans: 100
We know if 3 x 4 = 12 then :
12 ÷ 4 = 3
12 ÷ 3 = 4

9. If one of the number is 18 find the other number? Solution:
Example 4: The L. C. M. and H. C. F. of two numbers are 126 and 6 respectively.
If one of the number is 18 find the other number?
Solution:
Given:- LCM=126 , HCF=6 , Known Number = 18
To Find:- Unknown number
We know HCF x LCM = product of two numbers
126 x = ( x ? )
= x ?
756 ÷ = ?
So, Unknown number(?) = 42
Ans: 42
We know if 3 x 4 = 12 then :
12 ÷ 4 = 3
12 ÷ 3 = 4

10. Try these The LCM of 63 and 56 is 294. Find the HCF?
The H. C. F of 32 and 48 is 16. Find the L. C. M?
3. The L. C. M. and H. C. F. of two numbers 96 and 4 respectively.
If one of the number is 32 find the other number?
4. The L. C. M. and H. C. F. of two numbers 418 and 1 respectively.
If one of the number is 22 find the other number?