https://artofproblemsolving.com/wiki/index.php?title=2006_iTest_Problems/Problem_3&feed=atom&action=history2006 iTest Problems/Problem 3 - Revision history2024-03-28T12:56:26ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2006_iTest_Problems/Problem_3&diff=99111&oldid=prevRockmanex3: Solution to Problem 3 — products and optimization2018-11-27T06:30:44Z<p>Solution to Problem 3 — products and optimization</p>
<p><b>New page</b></p><div>==Problem==<br />
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Let <math>I, T, E, S</math> be distinct positive integers such that the product <math>ITEST = 2006</math>. What is the largest possible value of the sum <math>I + T + E + S + T + 2006</math>?<br />
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<math>\mathrm{(A)}\, 2086\quad\mathrm{(B)}\, 4012\quad\mathrm{(C)}\, 2144</math><br />
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==Solution==<br />
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Note that the expression can be written as <math>IT^2ES = 2006</math>. The prime factorization of <math>2006</math> is <math>2 \cdot 17 \cdot 59</math>. Since <math>2006</math> does not have perfect squares as factors, <math>T = 1</math>. The other variables are the three primes since all variables are distinct, so the maximum value of <math>I + T + E + S + T + 2006</math> is <math>2 + 2 + 17 + 59 + 2006 = \boxed{\textbf{(A) } 2086}</math>.<br />
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==See Also==<br />
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{{iTest box|year=2006|num-b=2|num-a=4|ver=[[2006 iTest Problems/Problem U1|U1]] '''•''' [[2006 iTest Problems/Problem U2|U2]] '''•''' [[2006 iTest Problems/Problem U3|U3]] '''•''' [[2006 iTest Problems/Problem U4|U4]] '''•''' [[2006 iTest Problems/Problem U5|U5]] '''•''' [[2006 iTest Problems/Problem U6|U6]] '''•''' [[2006 iTest Problems/Problem U7|U7]] '''•''' [[2006 iTest Problems/Problem U8|U8]] '''•''' [[2006 iTest Problems/Problem U9|U9]] '''•''' [[2006 iTest Problems/Problem U10|U10]]}}<br />
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[[Category:Introductory Number Theory Problems]]</div>Rockmanex3