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Find the difference between the smallest perfect square larger than one million and the largest perfect square smaller than one million.

I did not want to use a calculator for this question. I read this in a mathematical teaser book and was not sure how to solve it. I tried doing this with smaller numbers such as $10.$ The smallest perfect square larger than ten would would be $16$ and the smallest perfect square next to ten would be $9.$ I found the difference of those to be $7$ Then I tried this for $100.$ I found that the smallest perfect square is $81$ and the largest perfect square to be $121.$ I found the difference of those to be $40.$

However, I cannot seem to solve the one million question. Can someone help me to solve this? I was working on it for sometime now and would like to see how to solve it.

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    $\begingroup$ One million is the square of one thousand... $\endgroup$
    – lhf
    Oct 1, 2014 at 3:16

2 Answers 2

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We want $(x+1)^2-(x-1)^2$, where $x=1000$.

The difference is $4x$, that is, $4000$.

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  • $\begingroup$ Thank you. I get it now and I see how this works. I tried this with other problems like this and was able to solve this. $\endgroup$
    – col
    Oct 1, 2014 at 3:47
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    $\begingroup$ Good, I am happy that you got the general structural idea. $\endgroup$
    – André Nicolas
    Oct 1, 2014 at 3:51
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$a^2-b^2=(a-b)(a+b)$

Note that $100=10^2$ and thus the $a=11,b=9$ would be the values to plug into the formula above to get $(11-9)(11+9)=2*20=40$

Note that $1,000,000=1,000^2$ and using the same trick:

$(1001-999)(1001+999) = 2*2000 = 4000$ for another way to compute the value.

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  • $\begingroup$ +1 Thank you for you help. The explanation helped me to solve this. $\endgroup$
    – col
    Oct 1, 2014 at 3:47

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