if you place 9 in the left hand side of a 5-digit number, you get a 6-digit number.this 6-digit number is four times the 6-digit number that you get when you place 9 in the right hand side of the original 5 digit number. what is the sum of the digits of the original 5-digit number? all digits are distinctly different. what are they?

• Let the 5-digit number is : ABCDE ;[ Where A,B,C,D,E are th digits of the 5-digit number ].
  So, actually the 5-digit number is = (10^4)A +(10^3)B +(10^2)C + 10D + E
  
  Now, by the given condition:
  (9ABCDE)= 4*(ABCDE9)...[remember A,B,C,D,E are digits here and not multiplied]
  Or, in another way,we can write
  9*(10^5)+ (10^4)A +(10^3)B +(10^2)C + 10D + E =
  4*{(10^5)A+ (10^4)B +(10^3)C +(10^2)D + 10E + 9}
  Clearly A 9ABCD6 = 4*(ABCD69)....(eqn 1 )
  nOW,4*(69)=276
  so, the last 2-digits of 4*(ABCD69) will be (76).
  So, from (eqn 1 )
  we get 9ABCD6 = 4*(ABCD69)
  So,D=7
  => 9ABC76 = 4*(ABC769)............[eqn 2]
  Again we see that 4*(769)= 3076.
  So, last 3 digits of 4*(ABC769) will be : (076)
  So, we get from [eqn 2]
  9ABC76 = 4*(ABC769)
  where ,C=0
  => 9AB076 = 4*(AB0769)
  we see that 4*(0769)=
  last 4-digits of 4*(AB0769) will be (3076)
  So, a similar argument as above gives that
  from,
  9AB076 = 4*(AB0769)
  B=3
  =>9A3076=4*(A30769)
  Now, 4*(30769)=123076
  So,applying a similar argument we get,
  A=2 .
  So, the original 5-digit number is : ABCDE = 23076.
  
  To see whether it satisfies the condition or not, we observe that :
  923076 = 4*(230769), which is actually correct.
  
  So, original 5-digit number is : 23076 .
  
  THANKS to All.. :-):-) ..
• 11 years agoHelpfull: Yes(11) No(1)
• CORRECTION OF PRINTING ERROR :-
  ----------------------------------------------------
  in the above solution,
  plse consider the equation where I mentioned (eqn 1)
  there is a slight printing mistake, the (eqn 1) should be read as :
  
  Clearly , 9ABCD6 = 4*(ABCD69)....(eqn 1 )
  
  This is bcoz last digit of 4*(ABCDE9) will be =6 ; [since 4*9 = 36]
  so,the last digit of (9ABCDE) will be 6;
  So, E = 6...
  
  Now refer to the rest part of the solution..
  
  thanks to All..:-) :-)
• 11 years agoHelpfull: Yes(11) No(1)