indian1724
Pranjal committed a mistake in finding the LCM of three distinct positive integers greater than 1 namely A, B and C, and found it to be 840, which is a common multiple of A, B and C all, but is not the lowest. The HCF of A, B and C is 1. Find the maximum possible value of A + B + C.
1) 631
2) 613
3) 563
4) 257
The solution I thought is: 840 = 2*2*2*3*5*7
We are given 840 is not the LCM but a common multiple of A,B,C.
Therefore 840 is a multiple of the LCM.
As we need the maximum values for A,B,C we assume LCM to be 420.
If there was no condition of A,B,C being greater than 1, then we could simply assume
A=420, B=210, C=1 (satisfies the condition of HCF=1) giving us the sum of 631.
Here, I just put A the biggest, B second biggest and C as 1.
However as A,B,C are greater than 1, we need a prime number to be one of A,B,C.
okay so to maximize A,B,C, we can safely choose A = 420.
for B and C, one of them has to be prime and both of them have to such that they still satisfy our main constraint (LCM of A,B,C = 420), which means B,C have to be factors of 420.
so, to get factors of 420 that make the largest sum, we choose the smallest prime number (3) and 420/3 as our two numbers.
so finally we get A=420, B=140, C=3.
so, A+B+C=563
I know this isnt the best way to explain but i tried. thanks
we can’t take 420,210,2 as HCF wont be 1 anymore.