Q) Determine, with proof, the number of integer triples (x, y, z) satisfying
x^2 + 2y^2 + 98z^2 = 77777....7777 (2013 digits)

my solution :

working modulo ,



all perfect squares are

using all these cases - the only possibility is that

and

that means : and for some integers .

cancelling from both the sides - we have



that means that

but we can see that by euler's totient theorem :







so,

no integer solutions exist.