If a square number has a digit sum of 2019, that number is called a qualified number. How many qualified numbers are there?

(In reply to

Puzzle Answer by K Sengupta)

Suppose N be a qualified number.

Then, by divisibility rule of 9, we know that:

N=d(N)(mod 9), where d(N) is the digit sum of N

Now, in the given problem:

d(N)= 2019

=> d(N) = 3 (mod 9)

=> N= 3 (mod 9)

Now, 3 is NOT a quadratic residue in the mod 9 system.

This is a contradiction.

Consequently, there does NOT exist any qualified number and, the answer to the given question is 0.