2007 SMT Team Round Problem 6: Difference between revisions
Created page with "==Problem== <math>x\equiv\left(\sum_{k=1}^{2007} k \right) \mod 2016</math>, where <math>0\leq x\leq 2015</math>. Solve for <math>x</math>. ==Solution== The summation will gi..." |
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==Solution== | ==Solution== | ||
The summation will give us <math>1+2+3+\cdots+2006+2007=\frac{2007\times2008}2=2007\times1004</math>. Because <math>2007\equiv-9\mod 2016</math>, so <math>2007\times1004\equiv-9\times1004\equiv-9036\mod 2016</math>. So, we have <math>-9036\equiv-9036+2016\times5\equiv1044</math>, and because <math>0\leq1044\leq2016</math>, our answer is <math>\boxed{\mathrm{1044}}</math>. | The summation will give us <math>1+2+3+\cdots+2006+2007=\frac{2007\times2008}2=2007\times1004</math>. Because <math>2007\equiv-9\mod 2016</math>, so <math>2007\times1004\equiv-9\times1004\equiv-9036\mod 2016</math>. So, because we have <math>-9036\equiv-9036+2016\times5\equiv1044</math>, and because <math>0\leq1044\leq2016</math>, our answer is <math>\boxed{\mathrm{1044}}</math>. | ||
~Yuhao2012 | ~Yuhao2012 | ||
Latest revision as of 07:37, 28 October 2025
Problem
, where 
. Solve for 
.
Solution
The summation will give us 
. Because 
, so 
. So, because we have 
, and because 
, our answer is 
.
~Yuhao2012
