To find the remainder when $5^{50}$ is divided by 6 , we can look for a pattern in the remainders of powers of 5 when divided by 6 .
- $\(5^1=5\)$
$\(5 \div 6\)$ has a remainder of 5 .
(Alternatively, $\(5 \equiv-1(\bmod 6)\)$ )
- $\(5^2=25\)$
$\(25 \div 6=4\)$ with a remainder of 1 .
(Alternatively, $5^2 \equiv(-1)^2(\bmod 6) \equiv 1(\bmod 6)$ )
- $\(5^3=125\)$
$\(125 \div 6=20\)$ with a remainder of 5 .
(Alternatively, $\(5^3 \equiv 5^2 \cdot 5^1(\bmod 6) \equiv 1 \cdot 5(\bmod 6) \equiv 5(\bmod 6)\)$ )
The pattern of remainders when powers of 5 are divided by 6 is $\((5,1,5,1, \ldots)\)$.
The cycle length is 2 .
- If the exponent is odd, the remainder is 5 .
- If the exponent is even, the remainder is 1.

In this problem, the exponent is 50 , which is an even number.
Therefore, the remainder when $\(5^{50}\)$ is divided by 6 is $\(\mathbf{1}\)$.

Compare Quantity A and Quantity B:
Quantity A: The remainder, when $\(5^{50}\)$ is divided by 6 , is 1 .
Quantity B: 1
Since Quantity $A$ is 1 and Quantity $B$ is 1 , the two quantities are equal.
The final answer is The two quantities are equal.