Question:

Step 1: Express $x$ in terms of the given information. The $50^{\text{th}}$ root of a number $x$ is 12. This means that if we raise 12 to the power of 50, we get $x$. $$x^{1/50} = 12$$ Raising both sides to the power of 50, we get: $$x = 12^{50}$$ Step 2: Express $y$ in terms of the given information. The $50^{\text{th}}$ root of another number $y$ is 18. Similarly, we can express $y$ as 18 raised to the power of 50. $$y^{1/50} = 18$$ Raising both sides to the power of 50, we get: $$y = 18^{50}$$ Step 3: Calculate the sum $x+y$. Substitute the expressions for $x$ and $y$ found in the previous steps. $$x+y = 12^{50} + 18^{50}$$ Step 4: Find the remainder of $(x+y)$ when divided by 25. We need to find $(12^{50} + 18^{50}) \pmod{25}$. Let's evaluate each term modulo 25. For $12^{50} \pmod{25}$: We know that $12^2 = 144$. $144 = 5 \times 25 + 19$, so $144 \equiv 19 \pmod{25}$. Also, $19 \equiv -6 \pmod{25}$. So, $12^2 \equiv -6 \pmod{25}$. Then, $12^{50} = (12^2)^{25} \equiv (-6)^{25} \pmod{25}$. Since 25 is odd, $(-6)^{25} = -6^{25} \pmod{25}$. Now let's find $6^{25} \pmod{25}$: $6^1 \equiv 6 \pmod{25}$ $6^2 = 36 \equiv 11 \pmod{25}$ $6^3 \equiv 6 \times 11 = 66 \equiv 16 \pmod{25}$ $6^4 \equiv 11^2 = 121 \equiv 21 \pmod{25}$, or $6^4 \equiv -4 \pmod{25}$. $6^5 \equiv 6 \times (-4) = -24 \equiv 1 \pmod{25}$. So, $6^{25} = (6^5)^5 \equiv 1^5 \equiv 1 \pmod{25}$. Therefore, $12^{50} \equiv -1 \pmod{25}$. For $18^{50} \pmod{25}$: We know that $18 \equiv -7 \pmod{25}$. So, $18^{50} \equiv (-7)^{50} \pmod{25}$. Since the exponent 50 is even, $(-7)^{50} = 7^{50} \pmod{25}$. Now let's find $7^{50} \pmod{25}$: $7^2 = 49$. $49 = 1 \times 25 + 24$, so $49 \equiv 24 \pmod{25}$. Also, $24 \equiv -1 \pmod{25}$. So, $7^2 \equiv -1 \pmod{25}$. Then, $7^{50} = (7^2)^{25} \equiv (-1)^{25} \pmod{25}$. Since 25 is odd, $(-1)^{25} = -1$. Therefore, $18^{50} \equiv -1 \pmod{25}$. Finally, calculate the sum modulo 25: $$(x+y) \pmod{25} = (12^{50} + 18^{50}) \pmod{25}$$ $$\equiv (-1 + (-1)) \pmod{25}$$ $$\equiv -2 \pmod{25}$$ To express this as a positive remainder, we add 25: $$\equiv -2 + 25 \pmod{25}$$ $$\equiv 23 \pmod{25}$$ The remainder obtained on dividing $(x+y)$ by 25 is 23. The final answer is $\boxed{23}$.