Solution

\[ \begin{align*} \text{Let first number be} & = x\\ \text{Let other number be} & = 16x\\ \\ x \times 16 x & = 1296 \\ 16 x^2 & = 1296 \\ \\ x^2 & = \frac{\cancel{1296}^{81}}{\cancel{16}_1} \\ \\ x^2 & = 81 \\ x & = \sqrt{81} \\ x & = \sqrt{9 \times 9} \\ x & = 9 \\ \\ \color{green} \text{First number} & = \color{green} 9 \\ \\ \text{Other number} & = 16x \\ & = 16 \times 9 \\ \color{green} \text{Other number} & = \color{green} 144 \\ \\ \color{magenta} \text{Check} \implies 9 \times 144 &= \color{magenta}1496 \end{align*} \]

Answer The numbers are \( = \color{red} 9 \text{ and } 144 \)

Solution

\[ \begin{array}{r|l} & \phantom{0}225 \\ \hline 2 & \phantom{-}\overline{5} \, \overline{06} \, \overline{25} \\ & -4 \\ \hline 42 & \phantom{-}106 \\ & - \phantom{0}84 \\ \hline 445 & \phantom{-0}2225 \\ & -\phantom{0}2225 \\ \hline & \phantom{00000}0 \\ \end{array} \] \[ \begin{align*} \color{green} \sqrt{50625} &= \color{green} 225 \\ \\ \sqrt{506.25} + \sqrt{5.0625} & = \sqrt{\frac{50625}{100}} + \sqrt{\frac{50625}{10000}} \\ \\ & = \sqrt{\frac{(225)^2}{(10)^2}} + \sqrt{\frac{(225)^2}{(100)^2}} \\ \\ & = \sqrt{\left(\frac{225}{10}\right)^2} + \sqrt{\left(\frac{225}{100}\right)^2} \\ \\ & = \frac{225}{10} + \frac{225}{100} \\ \\ & = 22.5 + 2.25 \\ \color{green} \sqrt{506.25} + \sqrt{5.0625} & = \color{green} 24.75 \\ \end{align*} \]

Answer \( \color{red} \sqrt{50625} = 225 \) and \( \color{red} \sqrt{506.25} + \sqrt{5.0625} = 24.75 \)

Solution

\[ \begin{align*} \color{green} 2m &= \color{green} 14 \\ \\ m &= \frac{14}{2} \\ \\ m &= 7 \\ \\ m^2 - 1 &= 7^2 - 1 \\ & = 49 - 1\\ \color{green} m^2 - 1 &= \color{green} 48 \\ \\ m^2 + 1 &= 7^2 + 1 \\ &= 49 + 1 \\ \color{green} m^2 + 1 &= \color{green} 50 \\ \\ (2m, m^2 - 1, m^2 + 1) &= \color{green} (14,48,50) \end{align*} \]

Answer \( \color{red} (14,48,50) \) are the Pythagorean triplet.