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Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 10"

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==Problem==
 
==Problem==
 
Compute the [[remainder]] when <center><p><math>{2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}</math></p></center> is divided by 1000.
 
Compute the [[remainder]] when <center><p><math>{2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}</math></p></center> is divided by 1000.
 +
 
==Solution==
 
==Solution==
−
Since
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<!--By the [[binomial theorem]], note that
 +
 
 +
<cmath>\begin{eqnarray*} (1 + 1)^{2007} &=& {2007 \choose 0} + {2007 \choose 1} + \ldots + {2007 \choose 2007} = 2^{2007}\\
 +
(1 + \mathrm{cis} 120)^{2007} &=& {2007 \choose 0} + {2007 \choose 1} \mathrm{cis} 120 + \ldots + = -1 \\
 +
(1 + \mathrm{cis} 240)^{2007} &=& {2007 \choose 0} + {2007 \choose 1} \mathrm{cis} 240 + \ldots + = -1 \\
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\end{eqnarray*}</cmath>
 +
 
 +
If we add all three of these up, we get (since <math>\mathrm{cis} 120 + \mathrm{cis} 240 = -1</math>) the desired expression above. Hence we want
 +
-->
 +
Let <math>\omega</math> and <math>\zeta</math> be the two [[complex]] third-roots of 1.  Then let
 +
 
 +
<math>S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = \sum_{i = 0}^{2007} {2007 \choose i}(\omega^i + \zeta^i + 1)</math>.
 +
 
 +
Now, if <math>i</math> is a [[multiple]] of 3, <math>\omega^i + \zeta^i + 1 = 1 + 1 + 1 = 3</math>.  If <math>i</math> is one more than a multiple of 3, <math>\omega^i + \zeta^i + 1 = \omega + \zeta + 1 = 0</math>.  If <math>i</math> is two more than a multiple of 3, <math>\omega^i + \zeta^i + 1 = \omega^2 + \zeta^2 + 1= \zeta + \omega + 1 = 0</math>.  Thus
 +
 
 +
<math>S = \sum_{i = 0}^{669} 3 {2007 \choose 3i}</math>, which is exactly three times our desired expression.
 +
 
 +
We also have an alternative method for calculating <math>S</math>:  we know that <math>\{\omega, \zeta\} = \{-\frac{1}{2} + \frac{\sqrt 3}{2}i, -\frac{1}{2} - \frac{\sqrt 3}{2}i\}</math>, so <math>\{1 + \omega, 1 + \zeta\} = \{\frac{1}{2} + \frac{\sqrt 3}{2}i, \frac{1}{2} - \frac{\sqrt 3}{2}i\}</math>.  Note that these two numbers are both cube roots of -1, so <math>S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = (-1)^{669} + (-1)^{669} + 2^{2007} = 2^{2007} - 2</math>. 
 +
 
 +
Thus, the problem is reduced to calculating <math>2^{2007} - 2 \pmod{1000}</math>. <math>2^{2007} \equiv 0 \pmod{8}</math>, so we need to find <math>2^{2007} \pmod{125}</math> and then use the [[Chinese Remainder Theorem]]. Since <math>\phi (125) = 100</math>, by [[Euler's Totient Theorem]] <math>2^{20 \cdot 100 + 7} \equiv 2^7 \equiv 3 \pmod{125}</math>. Combining, we have <math>2^{2007} \equiv 128 \pmod{1000}</math>, and so <math>3S \equiv 128-2 \pmod{1000} \Rightarrow S\equiv \boxed{042}\pmod{1000}</math>.
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 +
==Solution 2 ==
 +
 
 +
<math>\sum_{k=0}^{n} {n \choose k} =2^n</math>, and <math>\sum_{k=0}^{3n} {3n \choose k} =2^{3n}</math>
 +
 
 +
<math>\sum_{k=0}^{3n} {3n \choose 3k}=\frac{\sum_{k=0}^{3n} {3n \choose k}+q(n)}{3}</math> where <math>q(n)</math> is an integer <math>-2 \le q(n) \le 2</math> that depends on the value of <math>n</math> and will make the sum an integer.  The division by 3 comes from the fact that we're skipping 2 out of every 3 terms in the binomial.  So we divide the whole sum by 3 and we add or subtract <math>q(n)</math> to correct for the integer based on the modularity of the sum with 3
 +
 
 +
<math>\sum_{k=0}^{3n} {3n \choose 3k}=\frac{2^{3n}+q(n)}{3}</math>
 +
 
 +
When <math>n</math> is odd, <math>3n</math> is odd, <math>2^{odd} \equiv (-1)^{odd}\;(mod\;3)\equiv -1\;(mod\;3)\equiv 2\;(mod\;3)</math>, Therefore <math>q(n)=-2</math> because we need to subract 2.
 +
 
 +
When <math>n</math> is even, <math>3n</math> is even, <math>2^{even} \equiv (1)^{even}\;(mod\;3)\equiv 1\;(mod\;3)\equiv -2\;(mod\;3)</math>, Therefore <math>q(n)=2</math> because we need to add 2.
 +
 
 +
So, the equation becomes:
 +
 
 +
<math>\sum_{k=0}^{3n} {3n \choose 3k}=\frac{2^{3n}+(-1)^n2}{3}</math>
  
 +
<math>{2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}=\sum_{k=0}^{3\times 669} { 3\times 669 \choose 3k }=\frac{2^{2007}+(-1)^{669}2}{3}=\frac{2^{2007}-2}{3}</math>
  
−
<math>\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}=2^n</math>
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<math>2^{2007} \equiv 2^7\;(mod\;1000)\equiv 128\;(mod\;1000)</math>
  
 +
<math>2^{2007}-2 \equiv 128-2\;(mod\;1000)\equiv 126\;(mod\;1000)</math>
  
−
that sum equals <math>2^{2007}</math>. So we just need to find n such that
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<math>\frac{2^{2007}-2}{3} \equiv \frac{126}{3}\;(mod\;1000)\equiv 42\;(mod\;1000)</math>
  
−
<math>n\equiv 2^{2007} \pmod {1000}</math>
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Therefore, the remainder when <math>{2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}</math> is divided by 1000 is <math>\boxed{042}</math>
  
 +
~Tomas Diaz. orders@tomasdiaz.com
  
 +
{{alternate solutions}}
  
−
{{solution}}
+
== See also ==
−
----
+
{{Mock AIME box|year=2006-2007|n=4|num-b=9|num-a=11|source=125025}}
  
−
*[[Mock AIME 4 2006-2007 Problems/Problem 11| Next Problem]]
+
[[Category:Intermediate Combinatorics Problems]]
−
*[[Mock AIME 4 2006-2007 Problems/Problem 9| Previous Problem]]
+
[[Category:Intermediate Number Theory Problems]]
−
*[[Mock AIME 4 2006-2007 Problems]]
 
−
* [[Binomial theorem]]
 
−
*[[Modular arithmetic]]
 

Latest revision as of 12:14, 25 November 2023

Problem

Compute the remainder when

${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}$

is divided by 1000.

Solution

Let $\omega$ and $\zeta$ be the two complex third-roots of 1. Then let

$S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = \sum_{i = 0}^{2007} {2007 \choose i}(\omega^i + \zeta^i + 1)$.

Now, if $i$ is a multiple of 3, $\omega^i + \zeta^i + 1 = 1 + 1 + 1 = 3$. If $i$ is one more than a multiple of 3, $\omega^i + \zeta^i + 1 = \omega + \zeta + 1 = 0$. If $i$ is two more than a multiple of 3, $\omega^i + \zeta^i + 1 = \omega^2 + \zeta^2 + 1= \zeta + \omega + 1 = 0$. Thus

$S = \sum_{i = 0}^{669} 3 {2007 \choose 3i}$, which is exactly three times our desired expression.

We also have an alternative method for calculating $S$: we know that $\{\omega, \zeta\} = \{-\frac{1}{2} + \frac{\sqrt 3}{2}i, -\frac{1}{2} - \frac{\sqrt 3}{2}i\}$, so $\{1 + \omega, 1 + \zeta\} = \{\frac{1}{2} + \frac{\sqrt 3}{2}i, \frac{1}{2} - \frac{\sqrt 3}{2}i\}$. Note that these two numbers are both cube roots of -1, so $S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = (-1)^{669} + (-1)^{669} + 2^{2007} = 2^{2007} - 2$.

Thus, the problem is reduced to calculating $2^{2007} - 2 \pmod{1000}$. $2^{2007} \equiv 0 \pmod{8}$, so we need to find $2^{2007} \pmod{125}$ and then use the Chinese Remainder Theorem. Since $\phi (125) = 100$, by Euler's Totient Theorem $2^{20 \cdot 100 + 7} \equiv 2^7 \equiv 3 \pmod{125}$. Combining, we have $2^{2007} \equiv 128 \pmod{1000}$, and so $3S \equiv 128-2 \pmod{1000} \Rightarrow S\equiv \boxed{042}\pmod{1000}$.

Solution 2

$\sum_{k=0}^{n} {n \choose k} =2^n$, and $\sum_{k=0}^{3n} {3n \choose k} =2^{3n}$

$\sum_{k=0}^{3n} {3n \choose 3k}=\frac{\sum_{k=0}^{3n} {3n \choose k}+q(n)}{3}$ where $q(n)$ is an integer $-2 \le q(n) \le 2$ that depends on the value of $n$ and will make the sum an integer. The division by 3 comes from the fact that we're skipping 2 out of every 3 terms in the binomial. So we divide the whole sum by 3 and we add or subtract $q(n)$ to correct for the integer based on the modularity of the sum with 3

$\sum_{k=0}^{3n} {3n \choose 3k}=\frac{2^{3n}+q(n)}{3}$

When $n$ is odd, $3n$ is odd, $2^{odd} \equiv (-1)^{odd}\;(mod\;3)\equiv -1\;(mod\;3)\equiv 2\;(mod\;3)$, Therefore $q(n)=-2$ because we need to subract 2.

When $n$ is even, $3n$ is even, $2^{even} \equiv (1)^{even}\;(mod\;3)\equiv 1\;(mod\;3)\equiv -2\;(mod\;3)$, Therefore $q(n)=2$ because we need to add 2.

So, the equation becomes:

$\sum_{k=0}^{3n} {3n \choose 3k}=\frac{2^{3n}+(-1)^n2}{3}$

${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}=\sum_{k=0}^{3\times 669} { 3\times 669 \choose 3k }=\frac{2^{2007}+(-1)^{669}2}{3}=\frac{2^{2007}-2}{3}$

$2^{2007} \equiv 2^7\;(mod\;1000)\equiv 128\;(mod\;1000)$

$2^{2007}-2 \equiv 128-2\;(mod\;1000)\equiv 126\;(mod\;1000)$

$\frac{2^{2007}-2}{3} \equiv \frac{126}{3}\;(mod\;1000)\equiv 42\;(mod\;1000)$

Therefore, the remainder when ${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}$ is divided by 1000 is $\boxed{042}$

~Tomas Diaz. orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See also

Mock AIME 4 2006-2007 (Problems, Source)
Preceded by
Problem 9 		Followed by
Problem 11
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