Hello, Swarnav!

A number whose sum of digits is 1977.

Prove that the number will not be a perfect square.

Note: in this problem, a "number" means a positive integer.

Consider thedigitalof a number.root

Add the digits together.

If the sum has more than one digit, add its digits together.

Continue until we have a one-digit sum.

That is the digital root of the number.

A number can have only nine possible digital roots.

Examine the digital roots of their squares.

. .

The digital root of a square must be:

Since the digital root of is , it cannotbe a square.