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Step-by-Step Solution
Step 1: Express x and y in exponential form
We are given that the 50th root of x is 12. Mathematically,
x^{\frac{1}{50}} = 12 , which implies
x = 12^{50}.
Similarly, the 50th root of y is 18. Hence,
y^{\frac{1}{50}} = 18 , which implies
y = 18^{50}.
Step 2: Understand the remainder calculation
We need to find the remainder upon dividing x + y = 12^{50} + 18^{50} by 25.
This is a problem in modular arithmetic: we want to evaluate
(12^{50} + 18^{50}) \bmod 25.
Step 3: Calculate 12^{50} \bmod 25
First, observe that 12 \equiv 12 \pmod{25} . Sometimes, it is helpful to
find a pattern or a smaller repeating cycle.
Compute 12^2 \bmod 25 :
12^2 = 144 \equiv 144 - 125 = 19 \pmod{25}.
Compute 12^3 \bmod 25 :
12^3 = 12 \times 12^2 = 12 \times 19 = 228 \equiv 228 - 225 = 3 \pmod{25}.
(Alternatively, one can express 3 \equiv -22 \pmod{25} , but we can keep track using 3.)
One can continue to find powers until identifying a pattern. By known or
worked out cycles, it can be shown that 12^{10} \equiv -1 \pmod{25}.
Thus, 12^{50} = (12^{10})^5 \equiv (-1)^5 = -1 \pmod{25}, which is the same as
24 \pmod{25}.
Step 4: Calculate 18^{50} \bmod 25
Notice 18 \equiv 18 \pmod{25}.
Compute 18^2 \bmod 25 :
18^2 = 324 \equiv 324 - 300 = 24 \equiv -1 \pmod{25}.
Hence 18^{50} = (18^2)^{25} \equiv (-1)^{25} = -1 \pmod{25}, which again is
24 \pmod{25}.
Step 5: Combine both results to find the final remainder
We now add these two results modulo 25:
\[
12^{50} + 18^{50} \equiv (-1) + (-1) = -2 \pmod{25}.
\]
Since -2 \equiv 23 \pmod{25}, the remainder when x+y is divided by 25 is 23.
Final Answer
The remainder obtained is 23.
