Is it possible to find?

Number Theory Level 4

1 x − 1 x y − 1 x y z = 19 97 \large \frac 1x - \frac1{xy} - \frac1{xyz} = \frac{19}{97}

Suppose we have positive integers x < y < z x<y<z such that it satisfy the equation above, evaluate x y z ‾ \overline{xyz} .

Details and Assumptions :

  • If x = 12 , y = 34 , z = 56 x=12,y=34,z=56 , then submit your answer as 123456.


The answer is 54997.

Naitik Sanghavi by naitik sanghavi , 20, India

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1 solution

Palash Som
Jan 30, 2016

HERE is a small hint for those who were unable to solve this problem by manipulating the above equation it can be written as follows 97y-97-19xy = 97/z since x , y , z are positive integers therefore the value of the above must be an integer too and since 97 is a prime number therefore z = 97 further manipulating the equation we get 97-19x = 97/y and since this also has an integral value so y can take the values 49 ,7 ,14 , 2. using trial and error we find it to be 49 and so x can also be found

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