To convert the number 4656 from base 10 to base 4, we follow these steps:

1. **Find the highest power of 4** that fits into 4656.
2. **Divide** the number by this power of 4 to find the coefficients.
3. **Subtract** the value represented by the coefficient and repeat for the next lower power of 4.

### Step-by-Step Conversion:

1. **Powers of 4**:  
   \( 4^0 = 1 \)  
   \( 4^1 = 4 \)  
   \( 4^2 = 16 \)  
   \( 4^3 = 64 \)  
   \( 4^4 = 256 \)  
   \( 4^5 = 1024 \)  
   \( 4^6 = 4096 \)  
   \( 4^7 = 16384 \)  (This is too large)

   The highest power of 4 less than 4656 is \( 4^6 = 4096 \).

2. **Divide 4656 by 4096** (i.e., \( 4^6 \)):
   \[
   4656 \div 4096 = 1 \quad \text{(quotient is 1, so the coefficient of \( 4^6 \) is 1)}
   \]
   Remaining value:  
   \[
   4656 - 4096 = 560
   \]

3. **Next, consider \( 4^5 = 1024 \)**:
   \[
   560 \div 1024 = 0 \quad \text{(quotient is 0, so the coefficient of \( 4^5 \) is 0)}
   \]
   Remaining value:  
   \[
   560 \text{ (unchanged since \( 1024 \times 0 = 0 \))}
   \]

4. **Next, consider \( 4^4 = 256 \)**:
   \[
   560 \div 256 = 2 \quad \text{(quotient is 2, so the coefficient of \( 4^4 \) is 2)}
   \]
   Remaining value:  
   \[
   560 - 512 = 48
   \]

5. **Next, consider \( 4^3 = 64 \)**:
   \[
   48 \div 64 = 0 \quad \text{(quotient is 0, so the coefficient of \( 4^3 \) is 0)}
   \]
   Remaining value:  
   \[
   48 \text{ (unchanged since \( 64 \times 0 = 0 \))}
   \]

6. **Next, consider \( 4^2 = 16 \)**:
   \[
   48 \div 16 = 3 \quad \text{(quotient is 3, so the coefficient of \( 4^2 \) is 3)}
   \]
   Remaining value:  
   \[
   48 - 48 = 0
   \]

7. **Next, consider \( 4^1 = 4 \)**:
   \[
   0 \div 4 = 0 \quad \text{(quotient is 0, so the coefficient of \( 4^1 \) is 0)}
   \]

8. **Finally, \( 4^0 = 1 \)**:
   \[
   0 \div 1 = 0 \quad \text{(quotient is 0, so the coefficient of \( 4^0 \) is 0)}
   \]

### Putting It All Together:
The base 4 representation of 4656 is **102300**.

### Final Answer:
\( 4656_{10} = 102300_4 \)

Would you like to ask anything more about this conversion or any other questions? Here are some related questions to consider:

1. How do you convert numbers from other bases to base 10?
2. What are some practical applications of different base systems?
3. How can we convert fractional numbers to different bases?
4. What is the process to convert a number from base 4 to base 8?
5. What is the significance of base 4 in computer science?
6. How do you add and subtract numbers in base 4?
7. How do you multiply and divide numbers in base 4?
8. What are some other examples of converting large numbers to base 4?

**Tip:** Practice converting numbers between different bases to strengthen your understanding of base systems!

Learn how to convert the number 4656 from base 10 to a numeral in base 4 with a detailed step-by-step solution. This problem covers key mathematical concepts including number bases and conversion methods. Suitable for students in Grades 7-9.

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