Given:
- When the positive integer $N$ is divided by 16 , the remainder is 3 .
- We want to find all possible remainders when $N$ is divided by 6 .

Step 1: Express $N$ mathematically
If $N$ divided by 16 leaves a remainder 3 , then:

$$
\(N=16 k+3 \quad \text { for some integer } k \geq 0\) .
$$


Step 2: Find the remainder of $N$ when divided by 6
We want the remainder when $N$ is divided by 6 .
Write:

$$
\(N=16 k+3\) .
$$


Take modulo 6 on both sides:

$$
\(N \quad \bmod 6=(16 k+3) \quad \bmod 6=(16 k \bmod 6)+(3 \bmod 6) \).
$$


Calculate each term modulo 6:
- $\(16 \bmod 6=4\)$,
so $16 k \bmod 6=4 k \bmod 6$.
- $\(3 \bmod 6=3\)$.

Thus:

$$
\(N \quad \bmod 6=(4 k \quad \bmod 6)+3 \).
$$

Step 3: Determine possible values of $\(4 k \bmod 6\)$
Calculate $\(4 k \bmod 6\)$ as $k$ varies:
- For $\(k=0,4 \times 0=0 \bmod 6=0\)$
- For $k=1,4 \times 1=4 \bmod 6=4$
- For $\(k=2,4 \times 2=8 \bmod 6=2\)$
- For $\(k=3,4 \times 3=12 \bmod 6=0\)$ (cycle repeats every 3)
- For $\(k=4,4 \times 4=16 \bmod 6=4\)$
- For $\(k=5,4 \times 5=20 \bmod 6=2\)$

So the possible values for $\(4 k \bmod 6\)$ are: $0,4,2$, cycling every 3 values of $k$.
Step 4: Add 3 to each possible value
Add the 3 from earlier:

$$
\(N \quad \bmod 6=(4 k \quad \bmod 6)+3 \quad \bmod 6\) .
$$


Calculate each:
- $\(0+3=3 \bmod 6=3\)$,
- $\(4+3=7 \bmod 6=1\)$,
- $\(2+3=5 \bmod 6=5\)$.

Step 5: Possible remainders are

$$
\(1,3,5 \).
$$


Answer selection:
- (A) O - No
- (B) $\(1-\mathrm{Yes}\)$
- (C) 2 - No
- (D) $\(3-\mathrm{Yes}\)$
- (E) 4 - No
- (F) $\(5-\mathrm{Yes}\)$

Final answer:
The possible remainders when $N$ is divided by 6 are 1,3 , and $\(5(\mathrm{~B}, \mathrm{D}, \mathrm{F})\)$.