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2007 iTest Problems/Problem 25: Difference between revisions

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Created page with "== Problem == Ted's favorite number is equal to <cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath> Find..."
 
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Solution to Problem 25
 
Line 1: Line 1:
== Problem ==
==Problem==


Ted's favorite number is equal to
Ted's favorite number is equal to
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath>
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath>


Find the remainder when Ted's favorite number is divided by 25.
Find the remainder when Ted’s favorite number is divided by 25.


<math>\text{(A) } 0\qquad
<math>\text{(A) } 0\qquad
Line 14: Line 14:
\text{(G) } 6\qquad
\text{(G) } 6\qquad
\text{(H) } 7\qquad
\text{(H) } 7\qquad
\text{(I) } 8\qquad \\ </math>
\text{(I) } 8\qquad</math>


<math>\text{(J) } 9\qquad
<math>\text{(J) } 9\qquad
Line 23: Line 23:
\text{(O) } 14\qquad
\text{(O) } 14\qquad
\text{(P) } 15\qquad
\text{(P) } 15\qquad
\text{(Q) } 16\qquad \\ </math>
\text{(Q) } 16\qquad</math>


<math>\text{(R) } 17\qquad
<math>\text{(R) } 17\qquad
Line 30: Line 30:
\text{(U) } 20\qquad
\text{(U) } 20\qquad
\text{(V) } 21\qquad
\text{(V) } 21\qquad
\text{(A) } 22\qquad
\text{(W) } 22\qquad
\text{(X) } 23\qquad
\text{(X) } 23\qquad
\text{(Y) } 24</math>
\text{(Y) } 24</math>


== Solution ==
==Solution==
 
Let <math>T = \sum_{i=1}^{2007} i \cdot \binom{2007}{i}</math>.  That means
<cmath>2T = 2 \cdot \sum_{i=1}^{2007} (i \cdot \binom{2007}{i}</cmath>
We know that <math>\binom{2007}{i} = \binom{2007}{2007-i}</math>, so
<cmath>2T = \sum_{i=1}^{2007} (i \cdot \binom{2007}{i}) + \sum_{j=1}^{2007} (j \cdot \binom{2007}{2007-j})</cmath>
If <math>2007-j= i</math>, then <math>i+j = 2007</math>, so
<cmath>2T = 2007 \cdot \sum_{i=0}^{2007} \binom{2007}{i}</cmath>
<cmath>2T = 2007 \cdot 2^{2007}</cmath>
<cmath>T = 2007 \cdot 2^{2006}</cmath>
Note that <math>2^{10} \equiv -1 \pmod{25}</math>.  Using the properties of [[modular arithmetic]], we can find the remainder when <math>T</math> is divided by <math>25</math>.
<cmath>T \equiv 2007 \cdot 2^{2000} \cdot 2^6 \pmod{25}</cmath>
<cmath>T \equiv 2007 \cdot (2^{10})^{200} \cdot 64 \pmod{25}</cmath>
<cmath>T \equiv 7 \cdot 1 \cdot 14 \pmod{25}</cmath>
<cmath>T \equiv 98 \pmod{25}</cmath>
<cmath>T \equiv 23 \pmod{25}</cmath>
The remainder when Ted’s favorite number is divided by 25 is <math>\boxed{\textbf{(X) } 23}</math>.
 
==See Also==
{{iTest box|year=2007|num-b=24|num-a=26}}
 
[[Category:Intermediate Number Theory Problems]]

Latest revision as of 13:35, 29 July 2018

Problem

Ted's favorite number is equal to \[1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}\]

Find the remainder when Ted’s favorite number is divided by 25.

$\text{(A) } 0\qquad \text{(B) } 1\qquad \text{(C) } 2\qquad \text{(D) } 3\qquad \text{(E) } 4\qquad \text{(F) } 5\qquad \text{(G) } 6\qquad \text{(H) } 7\qquad \text{(I) } 8\qquad$

$\text{(J) } 9\qquad \text{(K) } 10\qquad \text{(L) } 11\qquad \text{(M) } 12\qquad \text{(N) } 13\qquad \text{(O) } 14\qquad \text{(P) } 15\qquad \text{(Q) } 16\qquad$

$\text{(R) } 17\qquad \text{(S) } 18\qquad \text{(T) } 19\qquad \text{(U) } 20\qquad \text{(V) } 21\qquad \text{(W) } 22\qquad \text{(X) } 23\qquad \text{(Y) } 24$

Solution

Let $T = \sum_{i=1}^{2007} i \cdot \binom{2007}{i}$. That means \[2T = 2 \cdot \sum_{i=1}^{2007} (i \cdot \binom{2007}{i}\] We know that $\binom{2007}{i} = \binom{2007}{2007-i}$, so \[2T = \sum_{i=1}^{2007} (i \cdot \binom{2007}{i}) + \sum_{j=1}^{2007} (j \cdot \binom{2007}{2007-j})\] If $2007-j= i$, then $i+j = 2007$, so \[2T = 2007 \cdot \sum_{i=0}^{2007} \binom{2007}{i}\] \[2T = 2007 \cdot 2^{2007}\] \[T = 2007 \cdot 2^{2006}\] Note that $2^{10} \equiv -1 \pmod{25}$. Using the properties of modular arithmetic, we can find the remainder when $T$ is divided by $25$. \[T \equiv 2007 \cdot 2^{2000} \cdot 2^6 \pmod{25}\] \[T \equiv 2007 \cdot (2^{10})^{200} \cdot 64 \pmod{25}\] \[T \equiv 7 \cdot 1 \cdot 14 \pmod{25}\] \[T \equiv 98 \pmod{25}\] \[T \equiv 23 \pmod{25}\] The remainder when Ted’s favorite number is divided by 25 is $\boxed{\textbf{(X) } 23}$.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 24 		Followed by:
Problem 26
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