2007 SMT Team Round Problem 6

Problem

$x\equiv\left(\sum_{k=1}^{2007} k \right) \mod 2016$ , where $0\leq x\leq 2015$ . Solve for $x$ .

Solution

The summation will give us $1+2+3+\cdots+2006+2007=\frac{2007\times2008}2=2007\times1004$ . Because $2007\equiv-9\mod 2016$ , so $2007\times1004\equiv-9\times1004\equiv-9036\mod 2016$ . So, we have $-9036\equiv-9036+2016\times5\equiv1044$ , and because $0\leq1044\leq2016$ , our answer is $\boxed{\mathrm{1044}}$ .

~Yuhao2012

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2007 SMT Team Round Problem 6

Problem

Solution