x and y are two natural numbers such that x>y,x-y=66 and LCM(x,y)=360.Find x and y.
Ask Question , 7 Years ago
Grade 10
10 grade maths> x and y are two natural numbers such that...
1 Answers
Arun
Last Activity: 7 Years ago
Dear student
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
zn = 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)
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
zn = 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)
Regards
Arun (askIITians forum expert)
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