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.
Explanation to Puzzle Answer
