Let's break down and solve each problem:

1. Convert $110011_2$ to decimal:

$1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 16 + 0 + 0 + 2 + 1 = 51$ So, $110011_2 = 51_{10}$.

2. Convert $110011_2$ to octal:

First, group the binary number in sets of three from right to left: $110\ 011_2$ Convert each group to octal: $110_2 = 6_8 \quad \text{and} \quad 011_2 = 3_8$ So, $110011_2 = 63_8$.

3. Convert $345_{10}$ to binary:

Let's divide 345 by 2 repeatedly and note the remainders: $345 \div 2 = 172 \quad \text{remainder} \ 1$ $172 \div 2 = 86 \quad \text{remainder} \ 0$ $86 \div 2 = 43 \quad \text{remainder} \ 0$ $43 \div 2 = 21 \quad \text{remainder} \ 1$ $21 \div 2 = 10 \quad \text{remainder} \ 1$ $10 \div 2 = 5 \quad \text{remainder} \ 0$ $5 \div 2 = 2 \quad \text{remainder} \ 1$ $2 \div 2 = 1 \quad \text{remainder} \ 0$ $1 \div 2 = 0 \quad \text{remainder} \ 1$ So, reading the remainders from bottom to top, $345_{10} = 101011001_2$.

4. Convert $345_{10}$ to hexadecimal:

Let's divide 345 by 16: $345 \div 16 = 21 \quad \text{remainder} \ 9$ $21 \div 16 = 1 \quad \text{remainder} \ 5$ $1 \div 16 = 0 \quad \text{remainder} \ 1$ So, reading from bottom to top, $345_{10} = 159_{16}$.

5. Convert $767_8$ to decimal:

$7 \times 8^2 + 6 \times 8^1 + 7 \times 8^0 = 448 + 48 + 7 = 503$ So, $767_8 = 503_{10}$.

6. Convert $767_8$ to binary:

First, convert each octal digit to its binary equivalent: $7_8 = 111_2 \quad 6_8 = 110_2 \quad 7_8 = 111_2$ So, $767_8 = 111110111_2$.

7. Convert $9877_{16}$ to decimal:

$9 \times 16^3 + 8 \times 16^2 + 7 \times 16^1 + 7 \times 16^0 = 36864 + 2048 + 112 + 7 = 39031$ So, $9877_{16} = 39031_{10}$.

8. Convert $9877_{16}$ to octal:

First, convert $9877_{16}$ to binary: $9_{16} = 1001_2 \quad 8_{16} = 1000_2 \quad 7_{16} = 0111_2 \quad 7_{16} = 0111_2$ So, $9877_{16} = 1001100001110111_2$.

Next, group the binary number into sets of three from right to left and convert to octal: $100\ 110\ 000\ 111\ 011_2 = 4\ 6\ 0\ 7\ 3_8$ So, $9877_{16} = 46073_8$.

---

Would you like further explanations or details on any of these conversions?

Here are 5 related questions for further practice:

1. Convert $110011_2$ to hexadecimal.
2. Convert $101011001_2$ to octal.
3. Convert $159_{16}$ to binary.
4. Convert $503_{10}$ to binary.
5. Convert $9877_{16}$ to binary.

Tip: Always double-check each step in number base conversions, especially when dealing with different number systems, to avoid mistakes.