Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 1G (Number Theory)

For congruence classes in $\mathbb{Z}/N\mathbb{Z}$, defineThese operations are independent of representatives. If $a^{\prime }=a+Nr$ and $b^{\prime }=b+Ns$, then $(a^{\prime }+b^{\prime })-(a+b)=N(r+s)$ and $a^{\prime }{b}^{\prime }-ab=N(rb+sa+Nrs)$. Thus changing representatives leaves both results in the same congruence class. The usual integer laws descend to make this a commutative ring with zero $[0]$ and identity $[1]$.

The decimal expansion gives $M={\sum }_{i=0}^s{a}_{i}10^i$. In modular arithmetic, $10\equiv -1(\mathrm{mod}11)$, and henceAn integer is divisible by eleven exactly when its congruence class is zero, proving both directions of the alternating-digit test.