Type - 6
18 The product of the L.C.M and H.C.F of two
numbers is 24. The difference of two numbers is
2. Find the numbers :-
(1) 2 and 4
1) 6 and 4
(3) 8 and 6
(4) 8 and 10
Answers
Answer:
option a. ok. ok think and write
Second Option
Step-by-step explanation:
Given:-
The product of the L.C.M and H.C.F of two numbers is 24. The difference of two numbers is 2.
To find:-
Find the numbers ?
Solution:-
Let the required numbers be a and b
Given that
The product of the L.C.M and H.C.F of two
numbers is 24.
=> LCM × HCF = 24
We know that
The product of the LCM and the HCF is equal to the product of the two numbers
=> a× b = 24
ab = 24 ---------(1)
and
The difference of two numbers = 2
a - b = 2 --------(2)
We Know that
(a+b)^2 = (a-b)^2+4ab
=> (a+b)^2 = (2)^2+4(24)
=>(a+b)^2 = 4+96
=> (a+b)^2 = 100
=> a+b = ±√100
=> a+b = ±10
a+b = 10---------(3)
(since it can not be negative)
On adding (2)&(3)
a - b = 2
a + b = 10
(+)
__________
2a + 0 = 12
__________
=> 2a = 12
=>a = 12/2
=> a = 6
On Substituting the value of a in (2)
6-b = 2
=> b =6- 2
=> b = 4
Therefore a = 6 and b=4
Answer:-
The two numbers are 6 and 4
Check:-
a = 6 and b = 4
Their difference = a-b = 6-4=2
LCM of 6 and 4 = 12
HCF of 6 and 4 = 2
Their product = 12×2 = 24
Product of the numbers = 6×4 = 24
LCM × HCF = a×b
Verified the given relations.
Used formulae:-
- The product of the LCM and the HCF is equal to the product of the two numbers
- (a+b)^2 = (a-b)^2+4ab