The difference of two numbers is 1564. After dividing the larger number by the smaller, we get 6 as quotient and 19 as remainder. What is the smaller number?

$$\eqalign{ & {\text{Let two numbers}} = a{\text{ and }}b \cr & a > b \cr & a - b = 1564{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( {\text{i}} \right) \cr & 6b + 19 = a \cr & a - 6b = 19{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( {{\text{ii}}} \right) \cr & {\text{From equation }}\left( {\text{i}} \right){\text{ and }}\left( {{\text{ii}}} \right) \cr & a - b = 1564 \cr & a - 6b = 19 \cr & - \,\,\,\,\, + \,\,\,\,\, - \cr & \overline {\,\,\,\,\,\,\,5b = 1545\,} \cr & b = 309 \cr & {\text{Smaller number}} = 309 \cr} $$