Sachin9
Hi Karishma,
I cehcked the link but I did not understand why LCM will be a multiple of 10.. I understand that LCM*GCD=prod of 2 nos.. but don't understand why LCM has to be a multiple of 10..
Also, could you please help us with an approach on generating numbers for testing when a LCM is given.. in this case, lcm of x and y is 100.. .so how do we generate numbers whose lcm would be 100.
For gcd, we can write the GCD and multiply that by different numbers to get the pair of numbers..
Kindly explain how to do that for LCM
GIven that GCD = 10. What is GCD? It is the greatest common factor of two numbers i.e. both the numbers must have that factor. When you find the LCM of the numbers, the LCM includes all the factors of both the numbers. Hence, it will include 10 too.
e.g.
GCD = 10
Numbers: 10x, 10y where x and y are co-prime.
What will be the LCM?
LCM = 10xy (includes all factors of both the numbers)
In this question you don't need to list out the possible numbers given LCM = 100 but if you need to do it in another question, this is how you can handle that:
LCM = 100 = 2^2*5^2
Numbers:
Split the primes -> (4, 25)
Make one number = LCM -> (1, 100), (2, 100), (4, 100), (5, 100), (10, 100), (20, 100), (25, 100), (50, 100), (100, 100)
One number must have the highest power of each prime -> (2^2*5, 2*5^2 which is 20, 50), (2^2, 2*5^2 which is 4, 50), (2^2*5, 5^2 which is 20, 25)
The overall strategy is this: Split the LCM into its prime factors. At least one number must have the highest power of each prime.
LCM = 2^a*3^b*5^c
At least one number must have 2^a, same or another number must have 3^b and same or another number must have 5^c. There are various possibilities.