Difference between revisions of "Mock AIME 3 2006-2007 Problems/Problem 4"
(Created page with "== Problem == Let <math>x=(i-11)(i-10)....(i+8)(i+9)</math> where <math>i=\sqrt{-1}</math>. If <math>a</math> and <math>b</math> are positive integers such that <math>x(a-bi)</ma...") |
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== Solution == | == Solution == | ||
| + | Pairing the terms <math>(i-k)</math> with <math>(i+k)</math> for <math>k=1,2,...9</math> all result in integers, so we're left with making <math>i(i-10)(i-11)</math> an integer. | ||
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| + | <math>i(i-10)(i-11)=(21+109i)</math>. To make this an integer, we just multiply it by its conjugate <math>(21-109i)</math>, so <math>a+b=21+109=\boxed{130}</math> | ||
Latest revision as of 23:35, 15 February 2015
Problem
Let 
where 
. If 
and 
are positive integers such that 
is an integer, then find the minimum value of 
Solution
Pairing the terms 
with 
for 
all result in integers, so we're left with making 
an integer.
. To make this an integer, we just multiply it by its conjugate 
, so 
