Decimal expansion ($M={\sum }_{i=0}^s{a}_{i}10^i$)

A nonnegative integer's decimal expansion uses digits $0\le a_i\le 9$. Testing divisibility reduces this polynomial in ten modulo the divisor: for example $10\equiv -1(\mathrm{mod}11)$ makes divisibility by eleven equivalent to vanishing of the alternating digit sum.