Thread: Number theory

a number which sum of digit is 1977. Prove that the number will not be a perfect square

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 the digital root of a number.
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 cannot be a square.

but 6660 is not a perfect square, but it digit sum is 6+6+6+0=18,then 1+8=9.

Hello, Swarnav!

But 6660 is not a perfect square, but it digit sum is 6 + 6 + 6 + 0 = 18, then 1 + 8 = 9.

What is your point?


Read my post again . . .

I said: If a number is a square, then its digital root is 1, 4, 7 or 9.


You are taking the converse:
. . "If a number's digital root is 1, 4, 7 or 9, then it must be a square."
I didn't say that.


We cannot trust the converse.

Consider the true statement: "If the figure is a square, then it has four side."

The converse is: "If the figure has four sides, then it is a square."

This statement may or may not be true . . . agreed?

yes,i have understood. thanks. but please can you help me to find a general form for all valid numbers?