The solutions are x ≡ 1 mod 200 and y ≡ 5 mod 200.
System 1 (for x):
x ≡ 1 mod 8
x ≡ 2 mod 25
x ≡ 3 mod 81
Step 1: Find the Least Common Multiple (LCM) of the moduli:
LCM(8, 25, 81) = 200
Step 2: Multiply each congruence by its appropriate coefficient to make their moduli the LCM:
25x ≡ 25 mod 200 (multiply by 25 to make modulus 200)
8x ≡ 16 mod 200 (multiply by 8 to make modulus 200)
2x ≡ 6 mod 200 (multiply by 2 to make modulus 200)
Step 3: Add the three congruences:
65x ≡ 47 mod 200
Step 4: Solve for x modulo 200:
x ≡ 47 mod 200 (reduce by 65 mod 200)
Therefore, x ≡ 1 mod 200.
System 2 (for y):
y ≡ 5 mod 8
y ≡ 12 mod 25
y ≡ 47 mod 81
Repeat Steps 1-4 as above, finding the LCM (200) and multiplying each congruence to make their moduli 200.
Solution for y:
y ≡ 5 mod 200
Therefore, both x and y satisfy the respective systems of congruences when x ≡ 1 mod 200 and y ≡ 5 mod 200.
This means that x and y share the same residue when divided by 200, despite having different residues for their original moduli.