Let $q = \dfrac r s = \dfrac {1431818 \cdots 185} {1701818 \cdots 18560}$. Let the number of digits in $r$ be $n + 2$. Then the number of digits in $s$ is $n + 4$. By inspection, it is seen that $n$ is even. Then it is seen that: {{begin-eqn}} {{eqn | l = r       | r = 5 + 18 \times 10 + 18 \times 1000 + \cdots + 18 \times 10^{n - 5} + 18 \times 10^{n - 3} + 143 \times 10^{n - 1}       | c =  }} {{eqn | r = 5 + 10 \times \paren {18 + 18 \times 10^2 + \cdots + 18 \times 10^{n - 6} + 18 \times 10^{n - 4} } + 143 \times 10^{n - 1}       | c =  }} {{eqn | r = 5 + 10 \times \paren {18 \times \paren{10^2}^0 + 18 \times \paren{10^2}^1 + \cdots + 18 \times \paren{10^2}^{n / 2 - 3} + 18 \times \paren{10^2}^{n / 2 - 2} } + 143 \times 10^{n - 1}       | c =  }} {{eqn | r = 5 + 10 \paren {\sum_{i \mathop = 0}^{n / 2 - 2} 18 \times 10^{2 i} } + 143 \times 10^{n - 1}       | c =  }} {{eqn | r = 5 + 10 \times 18 \paren {\frac {10^{n - 2} - 1} {10^2 - 1} } + 143 \times 10^{n - 1}       | c = Sum of Geometric Sequence }} {{eqn | r = 5 \times \paren {10^2 - 1} + 10 \times 18 \paren {10^{n - 2} - 1} + 143 \times \paren {10^2 - 1} \times 10^{n - 1}       | c =  }} {{eqn | r = 5 \times 99 + 180 \times 10^{n - 2} - 180 + 143 \times 99 \times 10^{n - 1}       | c =  }} {{eqn | r = 5 \times 99 + 180 \times 10^{n - 2} - 180 + 143 \times 99 \times 10 \times 10^{n - 2}       | c =  }} {{eqn | r = 315 + 141750 \times 10^{n - 2}       | c =  }} {{eqn | r = 35 \times \paren {9 + 4050 \times 10^{n - 2} }       | c =  }} {{end-eqn}} and similarly: {{begin-eqn}} {{eqn | l = s       | r = 560 + 18 \times 1000 + 18 \times 100 \, 000 + \cdots + 18 \times 10^{3 + \paren {n - 3} } + 18 \times 10^{3 + \paren {n - 1} } + 170 \times 10^{n + 1}       | c =  }} {{eqn | r = 560 + 1000 \times \paren {18 + 18 \times 10^2 + \cdots + 18 \times 10^{n - 6} + 18 \times 10^{n - 4} } + 170 \times 10^{n + 1}       | c =  }} {{eqn | r = 560 + 1000 \times \paren {18 \times \paren {10^2}^0 + 18 \times \paren {10^2}^1 + \cdots + 18 \times \paren {10^2}^{n / 2 - 3} + 18 \times \paren {10^2}^{n / 2 - 2} } + 170 \times 10^{n + 1}       | c =  }} {{eqn | r = 560 + 1000 \paren {\sum_{i \mathop = 0}^{n / 2 - 2} 18 \times 10^{2 i} } + 170 \times 10^{n + 1}       | c =  }} {{eqn | r = 560 + 1000 \times 18 \paren {\frac {10^{n - 2} - 1} {10^2 - 1} } + 170 \times 10^{n + 1}       | c =  }} {{eqn | r = 560 \times \paren {10^2 - 1} + 1000 \times 18 \paren {10^{n - 2} - 1} + 170 \times \paren {10^2 - 1} \times 10^{n + 1}       | c =  }} {{eqn | r = 560 \times 99 + 18000 \times 10^{n - 2} - 18000 + 170 \times 99 \times 10^{n + 1}       | c =  }} {{eqn | r = 55440 + 18000 \times 10^{n - 2} - 18000 + 16830 \times 1000 \times 10^{n - 2}       | c =  }} {{eqn | r = 37440 + 16848000 \times 10^{n - 2}       | c =  }} {{eqn | r = 4160 \times \paren {9 + 4050 \times 10^{n - 2} }       | c =  }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = q = \dfrac r s       | r = \dfrac {35 \times \paren {9 + 4050 \times 10^{n - 2} } } {4160 \times \paren {9 + 4050 \times 10^{n - 2} } }       | c =  }} {{eqn | r = \dfrac {35 \times 41} {4160 \times 41}       | c =  }} {{eqn | r = \dfrac {1435} {170 \, 560}       | c =  }} {{end-eqn}} {{qed}}