(a) A 2-digit number is such that the product of its digits is 24. If 18 is subtracted from the number, the digits interchange their places. Find the…

Let the two numbers be \(\displaystyle x\) and \(\displaystyle y, \mathrm{x}>\mathrm{y}\) \[\begin{aligned} \therefore & x^{2}-y^{2}=180 \\ & y^{2}=8 x \end{aligned} \] Thus \(\displaystyle x^{2}-8 x-180=0\) \[\Rightarrow(x-18)(x+10)=0 \] Therefore, \(\displaystyle x=18\) \[\left.\begin{array}{r} \text { re, } x=18 \\ y= \pm 12 \end{array}\right\} \]