Thread: Problem Solving #1

Can an integer other than 1 with all its digits equal to 1 be the square of an
integer? Prove or disprove.

Well, 100 = 10^2, 10000 = 100^2,.....
S'that what you mean?

Originally Posted by matgrl

Can an integer other than 1 with all its digits equal to 1 be the square of an

integer? Prove or disprove.

So you're asking if any number of the form 11...1 can be a perfect square?

No - it can't be.

Hint - Look at the digit at the tenth place for such a number. It has to be even - which gives us a contradiction.

I guess....there should be some sort of way to solve this.

Could you further explain this idea...I am not sure I understand.

Originally Posted by aman_cc

No - it can't be.

Hint - Look at the digit at the tenth place for such a number. It has to be even - which gives us a contradiction.

Could you clarify this? The way I am reading it, it is not correct. For example, 16 is a perfect square, yet it has a 1 in the tens place.


One way to look at the problem is to consider all of the squares modulo 100. It's a little computationally intensive, but it will do the job. I computed the list in Mathematica:

{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 21, 44, 69, 96, 25, 56, 89, 24, 61, 0, 41, 84, 29, 76, 25, 76, 29, 84, 41, 0, 61, 24, 89, 56, 25, 96, 69, 44, 21, 0, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0}

We can see that 11 does not appear on the list. Therefore, any number of the form 11...1 (other than 1) cannot be a perfect square.

"Notice" that: 111111111111...111111111 is odd.

Now prove that 1111111...111111111 - 1 is not divided by 8 (why it will prove that any number of the from 11...1 (other than 1) cannot be a perfect square?)

What I meant is this.
For any perfect square to end in 1, say x = y^2. 'y' can only by of the form 10k + 1 or 10k - 1. This is easy to verify.
For either of these cases - the tenth place in 'x' will be even. This can be proved easily.

So, my claim is any perfect square ending in 1 will have an even number in it's tens place. And, number of the form 1111....111 violate that and hence can't be a perfect square.

@Also sprach Zarathustra - A very nice way to prove this. Thanks

Aman_cc I understand that I need to look at the tenths place...but would I do something like this:

9^2 = 81
29 ^2 = 841
11 ^2 = 121
21^2 = 441

I thought the question was asking if all numbers such as 1,11,111,1111,11111 be a square of an integer.


Could you possibly show me some examples of your thinking because I think you might be right.

I don't think you got my argument right.

Step 1: Prove that any perfect square with digit 1 at it's one's place will have an even digit at it's ten's place.

Can you do this using the hints I provided?

Step 2: A number of the form 11...11111 is clearly a voilation of the argument above, hence, can't be a perfect square.

This completes the proof. But I think the method by Also sprach Zarathustra is pretty neat/better