1. **State the problem:** Find the values of $a$ such that the distance between the points $(1, a)$ and $(3, 5)$ is $\sqrt{8}$.\n\n2. **Formula used:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:\n$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$\n\n3. **Apply the formula:** Substitute $(x_1, y_1) = (1, a)$ and $(x_2, y_2) = (3, 5)$ and $d = \sqrt{8}$:\n$$\sqrt{8} = \sqrt{(3 - 1)^2 + (5 - a)^2}$$\n\n4. **Simplify inside the square root:**\n$$(3 - 1)^2 = 2^2 = 4$$\nSo,\n$$\sqrt{8} = \sqrt{4 + (5 - a)^2}$$\n\n5. **Square both sides to eliminate the square root:**\n$$8 = 4 + (5 - a)^2$$\n\n6. **Isolate the squared term:**\n$$8 - 4 = (5 - a)^2$$\n$$4 = (5 - a)^2$$\n\n7. **Take the square root of both sides:**\n$$\sqrt{4} = \sqrt{(5 - a)^2}$$\n$$2 = |5 - a|$$\n\n8. **Solve the absolute value equation:**\n$$5 - a = 2 \quad \text{or} \quad 5 - a = -2$$\n\n9. **Find $a$ for each case:**\n- If $5 - a = 2$, then $a = 5 - 2 = 3$\n- If $5 - a = -2$, then $a = 5 + 2 = 7$\n\n**Final answer:**\n$$a = 3 \quad \text{or} \quad a = 7$$