The number $n$ is called a composite number if it can be written in the form $n = a\times b$, where $a, b$ are positive
integers greater than $1$. Write number $2013$ in a sum of $m$ composite numbers. What is the largest value of $m$?

(A): $500$, (B): $501$, (C): $502$, (D): $503$, (E): None of the above.
To solve the problem, we need to express the number 2013 as a sum of the maximum number of composite numbers. 

1. **Understanding Composite Numbers**:
   A composite number is a positive integer that has at least one positive divisor other than one or itself. In other words, it can be written as \( n = a \times b \) where \( a \) and \( b \) are integers greater than 1.

2. **Initial Decomposition**:
   We start by noting that 4 is the smallest composite number. We can use 4 to decompose 2013 into a sum of composite numbers. 

3. **Maximizing the Number of Terms**:
   To maximize the number of composite numbers in the sum, we should use the smallest composite number (which is 4) as many times as possible. 

4. **Calculation**:
   \[
   2013 = 4 \times m + r
   \]
   where \( m \) is the number of 4's we can use, and \( r \) is the remainder. We need \( r \) to be a composite number to ensure all parts of the sum are composite numbers.

5. **Finding \( m \) and \( r \)**:
   \[
   2013 \div 4 = 503 \text{ remainder } 1
   \]
   This means:
   \[
   2013 = 4 \times 503 + 1
   \]
   However, 1 is not a composite number. Therefore, we need to adjust the number of 4's to ensure the remainder is a composite number.

6. **Adjusting the Remainder**:
   We reduce the number of 4's by 1 and check the new remainder:
   \[
   2013 = 4 \times 502 + 5
   \]
   Here, 5 is not a composite number. We continue adjusting:
   \[
   2013 = 4 \times 501 + 9
   \]
   Here, 9 is a composite number (since \( 9 = 3 \times 3 \)).

7. **Conclusion**:
   The maximum number of composite numbers we can use is 501 terms of 4 and 1 term of 9, making a total of 502 composite numbers.

\[
\boxed{502}
\]