1. **Problem statement:** We are given two numbers whose arithmetic mean (AM) is 13 and geometric mean (GM) is 12. We need to find the two numbers and their harmonic mean (HM). 2. **Formulas:** - Arithmetic mean: $AM = \frac{x + y}{2}$ - Geometric mean: $GM = \sqrt{xy}$ - Harmonic mean: $HM = \frac{2xy}{x + y}$ 3. **Given:** - $AM = 13 \Rightarrow \frac{x + y}{2} = 13 \Rightarrow x + y = 26$ - $GM = 12 \Rightarrow \sqrt{xy} = 12 \Rightarrow xy = 144$ 4. **Find the two numbers $x$ and $y$:** We have the system: $$\begin{cases} x + y = 26 \\ xy = 144 \end{cases}$$ 5. **Solve for $x$ and $y$:** Consider the quadratic equation with roots $x$ and $y$: $$t^2 - (x + y)t + xy = 0 \Rightarrow t^2 - 26t + 144 = 0$$ 6. **Calculate the discriminant:** $$\Delta = 26^2 - 4 \times 1 \times 144 = 676 - 576 = 100$$ 7. **Find the roots:** $$t = \frac{26 \pm \sqrt{100}}{2} = \frac{26 \pm 10}{2}$$ So, - $t_1 = \frac{26 + 10}{2} = 18$ - $t_2 = \frac{26 - 10}{2} = 8$ Thus, the two numbers are $18$ and $8$. 8. **Find the harmonic mean:** $$HM = \frac{2xy}{x + y} = \frac{2 \times 18 \times 8}{26} = \frac{288}{26} = \frac{144}{13} \approx 11.08$$ **Final answer:** The two numbers are $18$ and $8$, and their harmonic mean is $\frac{144}{13}$ (approximately $11.08$).