1. ## Number theory problems

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

2. ## Re: Number theory problems

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.
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.

. . $\begin{array}{ccc}\text{digit root} && \text{digital root} \\ \text{of }n & n^2 & \text{of }n^2 \\ \hline 1 & 1 & 1 \\ 2 & 4 & 4 \\ 3 & 9 & 9 \\ 4 & 16 & 7 \\ 5 & 25 & 7 \\ 6 & 36 & 9 \\ 7 & 49 & 4 \\ 8 & 64 & 1 \\ 9 & 81 & 9 \end{array}$

The digital root of a square must be: $1, 4, 7,\text{ or }9.$

Since the digital root of $1977$ is $6$, it cannot be a square.

3. ## Re: Number theory problems

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

4. ## Re: Number theory problems

Hello, Swarnav!

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

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?

5. ## Re: Number theory problems

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