# Smallest number to get a perfect square

## Homework Statement

What is the smallest number than can be added to 2008 to get a perfect square?

## The Attempt at a Solution

I tried to factorise the number into its factors which in this case were 2*2*2*251. Then i saw that we needed another 2 and 251 to make the number a perfect square. But the number 502 has to multiplied to 2008 and not added to it, and also there are many more square numbers between 2008 and 2008*502

P.S. I would like the method too of obtaining the answer, in case you forget.

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diazona
Homework Helper
You could go with trial and error Add 1, see if that gets you a perfect square, if not add 2, see if that gets you a perfect square, etc. Although I'm sure you would figure out a shortcut before long.

Hint: you know that there are many square numbers between 2008 and 2008*502. Have you tried to figure out what the smallest of these is?

If i am reading the problem correctly, the square root of 2008 + some number should equal a whole number. The square root of 2008 is 44.811. The next whole number would be 45. The square root of what number when added to 2008 give you 45?

I think a perfect square can be expressed by X squared =A.
With your example X squared =2008+the number you wish to find
the root of 2008 is about 44.8 so round this up to the next nearest whole number and this gives X squared =49 squared=2401.The number,therefore=2401-2008=393

diazona
Homework Helper
Aren't we not supposed to give answers, as it's a homework question? (although 393 is not actually the right answer)

Aren't we not supposed to give answers, as it's a homework question? (although 393 is not actually the right answer)
Thanks diazona.Well, what a dopey dollop I am making 44+1 equal to 49
And I think you are probably right about it not being appropriate for this question to give an actual answer,albeit wrong.Sorry moderators ,genuine oversight.

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I don't know if this is quite the answer you were looking for, but what about adding negative 2008 so that you would arrive at 0?

Mentallic
Homework Helper
I don't know if this is quite the answer you were looking for, but what about adding negative 2008 so that you would arrive at 0?
lol that is if you consider "smallest number" to mean the largest negative :tongue:

All that is left is: $$45^2=2008+x$$ hence, $$x=45^2-2008$$

Can you take it from here?