1/x + 1/y = 1/z, if x =12 and y and z are whole numbers. find the no of solutions possible?

• AVANTIKA
  1/12 + 1/y = 1/z
  solving further you will get the equation 12-z=12 * z/y
  now z and y are whole numbers so the max value of z would be 12 on LHS .
  Form this we conclude that the range of z is from 1 to 12.
  
  Now solving the equation further you will get y = 12*z/12-z
  for every z you will get corresponding y . If the corresponding y is a whole no then that will be the solution . Counting all of them for every z = 3,4,6,8,9,10,11 we get corresponding y= calculate from formula
  And these 7 combination of z and y are solutions ..
• 7 years agoHelpfull: Yes(5) No(10)
• 1/12 + 1/y = 1/z
  (y + 12)/12y = 1/z
  z(y + 12) = 12y
  z(y + 12) - 12y =0
  z(y + 12) - 12y -144= -144
  z(y + 12) – 12(y -12)= -144
  (z-12)(y + 12) = -144
  (12-z)(y+12) = 144 = 2^4*3^2
  Total number of +ve solution = (4+1)(2+1) = 15
  Total Integral Solution = 2*15-1 = 29
  
  
  Generalization a/x-b/y= 1/n,
  1) Number of +ve solution = number of factors of (a*b*n^2)
  2) Number of Integral solution = 2* number of factors of (a*b*n^2) -1
• 7 years agoHelpfull: Yes(3) No(8)
• 7 solution
• 7 years agoHelpfull: Yes(1) No(3)
• Amit can you please explain?
• 7 years agoHelpfull: Yes(0) No(1)