How many positive four-digit integers n have the property that the three-digit number obtained by removing the leftmost digit is one-ninth of n?
+26367 How many positive four-digit integers n have the property that the three-digit number obtained by removing
the leftmost digit is one-ninth of n?
\(\begin{array}{|lrcll|} \hline (1) & n_4 &=& 10^3a + n_3 \\ \\ \hline \\ (2) & n_3 &=& \dfrac{n_4}{9} \\\\ & 9n_3 &=& n_4 \\\\ & 9n_3 &=& 10^3a + n_3 \\\\ & 8n_3 &=& 10^3a \quad & | \quad : 8 \\\\ & \mathbf{n_3} & \mathbf{=} & \mathbf{125a} \\ \hline \end{array} \)
\(\begin{array}{|r|c|l|l|} \hline & n_3=125a & \\ a & n_3\lt 1000 & n_4 &\\ \hline 1 & 125 & 1125 & \frac{1125}{9} = 125 \\ 2 & 250 & 2250 & \frac{2250}{9} = 250 \\ 3 & 375 & 3375 & \frac{3375}{9} = 375 \\ 4 & 500 & 4500 & \frac{4500}{9} = 500 \\ 5 & 625 & 5625 & \frac{5625}{9} = 625 \\ 6 & 750 & 6750 & \frac{6750}{9} = 750 \\ 7 & 875 & 7875 & \frac{7875}{9} = 875 \\ \hline \end{array}\)
+26367 How many positive four-digit integers n have the property that the three-digit number obtained by removing
the leftmost digit is one-ninth of n?
\(\begin{array}{|lrcll|} \hline (1) & n_4 &=& 10^3a + n_3 \\ \\ \hline \\ (2) & n_3 &=& \dfrac{n_4}{9} \\\\ & 9n_3 &=& n_4 \\\\ & 9n_3 &=& 10^3a + n_3 \\\\ & 8n_3 &=& 10^3a \quad & | \quad : 8 \\\\ & \mathbf{n_3} & \mathbf{=} & \mathbf{125a} \\ \hline \end{array} \)
\(\begin{array}{|r|c|l|l|} \hline & n_3=125a & \\ a & n_3\lt 1000 & n_4 &\\ \hline 1 & 125 & 1125 & \frac{1125}{9} = 125 \\ 2 & 250 & 2250 & \frac{2250}{9} = 250 \\ 3 & 375 & 3375 & \frac{3375}{9} = 375 \\ 4 & 500 & 4500 & \frac{4500}{9} = 500 \\ 5 & 625 & 5625 & \frac{5625}{9} = 625 \\ 6 & 750 & 6750 & \frac{6750}{9} = 750 \\ 7 & 875 & 7875 & \frac{7875}{9} = 875 \\ \hline \end{array}\)
