Let the GCD be x
then 2 numbers are mx,nx: with out loss of generality let m > n
now LCM = mnx ( as m and n are copimes)
so difference = (m-n) x = 66
mnx = 360
now 11 does not devide x so 11 devides (m-n)
m-n = 11 => x = 6 so mn = 60
m-n = 11
mn = 60
can be solved to give m = 15, n= 4 so numbers are 90 and 24
if m-n = 22 then x= 3 and
so mn = 120 does not have integer solution
m-n = 33 => x = 2 so mn = 180 does not have integer solution
m-n = 66 => x = 1 so mn = 360 does not have integer solution
so only solution (90,24)
13 comments:
now 11 does not devide x so 11 devides (m-n)
How can you say this . Please explain me.
as (m-n) x = 66 and 11 divides 66 so it must divide x or m-n. so as 11 does not devide x so 11 divides (m-n)
how can u say 11 does not divide x????
as product mnx is 360 so 11 does not divide product of m , n and x and hence it does not divide x because if 11 divides x then it divides mnx or360.
mister kali prasad , i am not getting one thing . yes i have understood till 11 divides (m-n). but i am not getting how
(m-n) = 11 and x = 6 . can u tell me the steps to get that , ( m-n) = 6 or how did you get that (m-n) = 6
how is (m-n) = 11 and x = 6 . please tell me the method to get that i have understood till the above step , but i am not getting how (m-n)=11 and x=6 . please tell me the method how did you get that ?
11 does not divide x because x divides 360 so 11 divides n-n. I never mentioned m -n = 6. what I mentioned is m-n is multiple of 11
Why 11 has come into picture first?
11 comes because it is factor of 66.
11 is not factor of mnx(which is 360) so not a factor of x
but it is a factor of (m-n) x = 66 so factor of m-n
need to be simplified further
it cannot be simplified further
I cannot understand this solution
which part you cannot understand
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