Monday, August 13, 2012

What are the two natural no. whose difference is 66 and the least common multiple is 360?

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)
at

13 comments:

  1. UnknownSeptember 3, 2013 at 7:47 PM

    now 11 does not devide x so 11 devides (m-n)
    How can you say this . Please explain me.

    ReplyDelete
  2. kaliprasad September 6, 2013 at 8:13 PM

    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)

    ReplyDelete
  3. UnknownOctober 8, 2013 at 2:36 PM

    how can u say 11 does not divide x????

    ReplyDelete
  4. kaliprasad October 10, 2013 at 8:13 PM

    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.

    ReplyDelete
  5. UnknownJune 6, 2015 at 3:02 PM

    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

    ReplyDelete
  6. UnknownJune 6, 2015 at 3:06 PM

    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 ?

    ReplyDelete
  7. kaliprasad June 21, 2015 at 9:51 AM

    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

    ReplyDelete
  8. UnknownJune 24, 2017 at 6:34 PM

    Why 11 has come into picture first?

    ReplyDelete
  9. kaliprasad June 26, 2017 at 10:48 AM

    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

    ReplyDelete
  10. UnknownOctober 18, 2017 at 8:56 PM

    need to be simplified further

    ReplyDelete
  11. kaliprasad October 19, 2017 at 5:15 PM

    it cannot be simplified further

    ReplyDelete
  12. UnknownNovember 24, 2019 at 12:11 PM

    I cannot understand this solution

    ReplyDelete
  13. kaliprasad November 26, 2019 at 11:54 AM

    which part you cannot understand

    ReplyDelete