https://artofproblemsolving.com/wiki/index.php?title=2008_iTest_Problems/Problem_61&feed=atom&action=history2008 iTest Problems/Problem 61 - Revision history2024-03-29T07:08:31ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2008_iTest_Problems/Problem_61&diff=96830&oldid=prevRockmanex3: Solution to Problem 61 (credit to official solution) -- harder units digit problem2018-08-07T17:54:02Z<p>Solution to Problem 61 (credit to official solution) -- harder units digit problem</p>
<p><b>New page</b></p><div>==Problem==<br />
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Find the units digits in the decimal expansion of <math>\left(2008+\sqrt{4032000}\right)^{2000}+\left(2008+\sqrt{4032000}\right)^{2001}+\left(2008+\sqrt{4032000}\right)^{2002}+\cdots</math><br />
<math>+\left(2008+\sqrt{4032000}\right)^{2007}+\left(2008+\sqrt{4032000}\right)^{2008}</math><br />
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==Solution==<br />
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Let <math>n</math> be a positive integer. Note that <math>2008^2 = 4032064,</math> which is close to <math>4032000.</math> That means <math>(2008-\sqrt{4032000})^{n}</math> is close to zero. With this in mind, we find that <math>(2008+\sqrt{4032000})^{n} + (2008-\sqrt{4032000})^{n}</math> is an integer, and the expansion is equal to <math>\sum_{i=2000}^{2008} \left( (2008+\sqrt{4032000})^{i} + (2008-\sqrt{4032000})^{i} \right) - (2008-\sqrt{4032000})^{i}.</math><br />
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To find that the units digit of <math>(2008+\sqrt{4032000})^{n} + (2008-\sqrt{4032000})^{n}</math>, note that in the expansion of <math>(2008+\sqrt{4032000})^{n}</math> and <math>(2008-\sqrt{4032000})^{n},</math> most of the terms end in a <math>0.</math> That means the units digit equals the units digit of <math>8^n + 8^n = 2 \cdot 8^n.</math><br />
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Now we need to find out how small <math>2008-\sqrt{4032000}</math> is. With a calculator, we find that <math>2008-\sqrt{4032000}=0.016 \text{.}</math> But even without a calculator, we can use inequalities to show that <math>2008-\sqrt{4032000} < \tfrac{1}{10}.</math> This means that <math>\sum_{i=2000}^{2008} (2008-\sqrt{4032000})^i</math> would be close to <math>0.</math><br />
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Thus, the units digit of the decimal expansion is <math>2+6+8+4+2+6+8+4+2-1 \rightarrow \boxed{1}.</math><br />
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==See Also==<br />
{{2008 iTest box|num-b=60|num-a=62}}<br />
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[[Category:Intermediate Number Theory Problems]]</div>Rockmanex3