Find all integral solutions to the equation
1/x - 1/xy - 1/xyz = 19/97.

Here it is not sufficent to find the number of solutions; you have to find all integral solutions and the only way I know that is by using Euclid's division lemma/division algorithm but I am not able to use that here because the RHS is a fraction. I am not completely exhausted with this problem. I am still trying it but I would like you to give me your solution too, as I can compare the two. But I have not given up yet.

Thanks a lot .
Yours sincerely,
Niranjan.

OK, similar idea to here . In the current form, the equations aren't very helpful - it would be much better if we could get rid of those fractions - because once we have an equation in integers we can often solve it by comparing prime factors, etc.

So of course we multiply through by 97xyz.

97yz - 97z - 97 = 19xyz

Now re-arrange this:

97yz - 97z - 19xyz = 97
z(97y - 97 - 19xy) = 97

This means that z is a factor of 97. There are only four integral factors of 97 as it is prime; namely -97,-1,1,97. So z = -97,-1,1 or 97.

• If z = -97:
  
  97y - 97 - 19xy = -1
  97y - 19xy = 96
  y(97 - 19x) = 96
  
  So 97 - 19x divides into 96. So x has to be positive (otherwise 97 - 19x > 96). Then 97 - 19x can take the following values with x = 1,2,3,...,10:
  
  78,59,40,21,2,-17,-36,-55,-74,-93, and clearly anything else smaller than -93 can't go into 96 an integral no. of times. The only one of these which is a factor of 96 is 2, which corresponds to x = 5. So if z = -97 then x = 5 and y = 48. So (5,48,-97) is a solution.
  
  
• If z = -1:
  
  97y - 97 - 19xy = -97
  
  97y = 19xy
  
  Nw we aren't allowed y = 0 otherwise your fractions would be undefined so:
  
  19x = 97
  
  There are no integral solutions to this as 19 doesn't go into 97 a whole no. of times. So z = -1 is not possible.
  
  
• If z = 1
  
  97y - 97 - 19xy = 97
  
  97(y - 2) = 19xy
  
  Now 19xy is a multiple of 19, so one of x,y is 97 or -97. If x = -97 then y - 2 = -19y, with no solution. If x = 97 then y - 2 = 19y with no solution. If y = -97 then y - 2 = -19x, -97 = -19x with no solution. If y = 97 then y - 2 = 19x, which gives x = 5. So the only solution here is (5,97,1)
  
  
• If z = 97
  
  Then
  
  97y - 97 - 19xy = 1
  
  y(97 - 19x) = 98
  
  So 97 - 19x goes into 98. If you let x run through 0,1,2,... you find 97 - 19x can take the following values (like before):
  
  97,78,59,40,21,2,-17,-36,-55,-74,-93 and again nothing smaller will work. So again the only possible value is 2 giving x = 5 and y = 49. So (5,49,97) is the final solution.