In basic medium manganese dioxide is oxidised by chlorate: $\mathrm{MnO}{}_2+\mathrm{ClO}{}_3{}^{-}+\mathrm{OH}{}^{-}\to \mathrm{MnO}{}_4{}^{2-}+\mathrm{Cl}{}^{-}+\mathrm{H}{}_2\mathrm{O}$. The sum of all coefficients (smallest whole numbers, reactants and products) is

Mn goes +4 to +6 (loses 2) and Cl goes +5 to -1 (gains 6), so $\mathrm{MnO}{}_2$ : $\mathrm{ClO}{}_3{}^{-}$ = 3 : 1. Charge balance (right side -7) needs 6 $\mathrm{OH}{}^{-}$ on the left, and then 3 $\mathrm{H}{}_2\mathrm{O}$ balance O and H: $3\mathrm{MnO}{}_2+\mathrm{ClO}{}_3{}^{-}+6\mathrm{OH}{}^{-}\to 3\mathrm{MnO}{}_4{}^{2-}+\mathrm{Cl}{}^{-}+3\mathrm{H}{}_2\mathrm{O}$. The coefficients add up to 17.