Quantity A: Remainder, when $2^N$ is divided by 10
To find the remainder when $2^N$ is divided by 10 , we need to look at the units digit of $\(2^N\)$.
Let's list the units digits of powers of 2 :
- $\(2^1=2\)$
- $\(2^2=4\)$
- $\(2^3=8\)$
- $\(2^4=16 \Longrightarrow 6\)$
- $\(2^5=32 \Longrightarrow 2\)$

The cycle of units digits for powers of 2 is ( $2,4,8,6$ ), which has a length of 4.

Since N is a positive integer, let's consider the possible values for N and their corresponding remainders:
- If $N=1$, remainder of $\(2^1 / 10\)$ is 2 .
- If $N=2$, remainder of $\(2^2 / 10\)$ is 4 .
- If $N=3$, remainder of $\(2^3 / 10\)$ is 8 .
- If $N=4$, remainder of $\(2^4 / 10\)$ is 6 .
- If $N=5$, remainder of $\(2^5 / 10\)$ is 2 .

The remainder when $\(2^N\)$ is divided by 10 will always be one of $(2,4,8,6)$.
This means the remainder for Quantity A can vary.
Quantity B: Remainder, when $\(7^N\)$ is divided by 5
To find the remainder when $\(7^N\)$ is divided by 5 , we can look at the units digit of $7^N$ and then divide by 5 , or find the remainders of powers of 7 when divided by 5 directly.

Let's find the remainders of powers of 7 when divided by 5 :
- $\(7^1(\bmod 5)=2(\bmod 5)\)$
- $\(7^2(\bmod 5)=49(\bmod 5)=4(\bmod 5)\)$
- $\(7^3(\bmod 5)=\left(7^2 \times 7^1\right)(\bmod 5)=(4 \times 2)(\bmod 5)=8(\bmod 5)=3(\bmod 5)\)$
- $\(7^4(\bmod 5)=\left(7^3 \times 7^1\right)(\bmod 5)=(3 \times 2)(\bmod 5)=6(\bmod 5)=1(\bmod 5)\)$
- $7\(^5(\bmod 5)=\left(7^4 \times 7^1\right)(\bmod 5)=(1 \times 2)(\bmod 5)=2(\bmod 5)\)$ The cycle of remainders for $\(7^N(\bmod 5)\)$ is $(2,4,3,1)$, which has a length of 4 .

The remainder when $7^N$ is divided by 5 will always be one of $(2,4,3,1)$.
This means the remainder for Quantity B can also vary.
Comparison:
Since both quantities depend on $N$ and can take on multiple values, we need to consider specific values of $N$.
- If $\mathrm{N}=1$ :
- Quantity A: Remainder of $\(2^1 / 10=2\)$.
- Quantity B: Remainder of $\(7^1 / 5=2\)$.
- In this case, Quantity A = Quantity B.
- If $\(\mathrm{N}=2\)$ :
- Quantity A: Remainder of $\(2^2 / 10=4\)$.
- Quantity B: Remainder of $\(7^2 / 5=49 / 5=4\)$.
- In this case, Quantity A = Quantity B.

- If $N=3$ :
- Quantity A: Remainder of $\(2^3 / 10=8\)$.
- Quantity B: Remainder of $\(7^3 / 5=343 / 5=3\)$.
- In this case, Quantity A is greater than Quantity B.
- If $\(\mathrm{N}=4\)$ :
- Quantity A: Remainder of $\(2^4 / 10=6\)$.
- Quantity B: Remainder of $\(7^4 / 5=2401 / 5=1\)$.
- In this case, Quantity A is greater than Quantity B.

Since the relationship between Quantity A and Quantity B changes depending on the value of $N$, we cannot determine a consistent relationship.

The final answer is The relationship cannot be determined from the information given.