Find the LCM and HCF of the following pairs of integers and verify that LCM x HCF = product of the integers:
(i)26 and 91
(ii)510 and 92
(iii)336 and 54
Answers
SOLUTION :
(i) Given :First number : 26
Second number : 91
The prime factors of 26 and 91 are :
26 = 2 x 13
91= 7 x 13
L.C.M (26,91) = 2 x 7 x 13
L.C.M of (26, 91) =182
[LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with highest power.]
H.C.F of (26, 91) = 13
[HCF(Highest Common Factor) of two or more numbers= Product of the smallest Power of each common prime factor involved in the numbers.]
We know that, L.C.M x H.C.F = First number x Second number
182 x 13 = 26 x 91
2366 = 2366
Hence verified
(ii) Given :First number : 510
Second number : 92
The prime factors of 510 and 92 are :
510 = 2 x 3 x 5 x 17
92 = 2 x 2 x 23 = 2² × 23
L.C.M ( 510 ,92) = 2 x 2 x 3 x 5 x 23 x 17
L.C.M (510 ,92) = 23460
[LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with highest power.]
H.C.F (510 , 92) = 2
[HCF(Highest Common Factor) of two or more numbers= Product of the smallest Power of each common prime factor involved in the numbers.]
We know that, L.C.M x H.C.F = First Number x Second Number
23460 x 2 = 510 x 92
46920 = 46920
Hence verified.
(iii) Given : First number : 336
Second number : 54
The prime factors of 336 and 54 are :
336 = 2 x 2 x 2 x 2 x 3 x 7 = 2⁴ × 3¹ × 7¹
54 = 2 x 3 x 3 x 3 = 2¹ × 3³
L.C.M (336 , 54) = 2⁴ × 3³× 7 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 7
L.C.M( 336 , 54) = 3024
[LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with highest power.]
H.C.F (336 , 54) = 2 × 3
H.C.F (336 , 54) = 6
[HCF(Highest Common Factor) of two or more numbers = Product of the smallest Power of each common prime factor involved in the numbers.]
We know that, L.C.M x H.C.F = First Number x Second Number
3024 x 6 = 336 x 54
18144 = 18144
Hence verified
HOPE THIS ANSWER WILL HELP YOU...
Q: Find the LCM and HCF of the following pairs of integers and verify that LCM x HCF = product of the integers :-
(i) 26 and 91
The prime factor of 26 and 91 :-
26 = 2 x 13
91= 7 x 13
LCM of 26 and 91 = 2 x 7 x 13 = 182
HCF of 26 and 91 = 13
Verification :-
LCM x HCF = 26 x 91 = 2366
Product of integers = 182 x 13 = 2366
LCM x HCF = product of the integers. Hence, verified
(ii) 510 and 92
The prime factor of 510 and 92 :-
510 = 2 x 3 x 5 x 17
92 = 2 x 2 x 23
LCM of 510 and 92 = 2 x 2 x 3 x 5 x 17 x 23 = 23460
HCF of 510 and 92 = 2
Verification :-
LCM x HCF = 510 x 92 = 46920
Product of integers = 23460 x 2 = 46920
LCM x HCF = product of the integers. Hence, verified
(iii) 336 and 54
The prime factor of 336 and 54 :-
336 = 2 x 2 x 2 x 2 x 3 x 7
54 = 2 x 3 x 3 x 3
LCM of 336 and 54 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 7 = 3024
HCF of 336 and 54 = 6
Verification :-
LCM x HCF = 336 x 54 = 18144
Product of integers = 3024 x 6 = 18144
LCM x HCF = product of the integers. Hence, verified