To solve the problem, we need to find the matrix \( X^6 \) given the matrix \( X = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} \), and then compute the expression \( a + b + 2020c + d \) where \( X^6 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
### Step 1: Calculate \( X^2 \)
To find \( X^6 \), we first calculate \( X^2 \):
\[
X^2 = X \cdot X = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}
\]
Calculating the elements:
- First row, first column: \( 2 \cdot 2 + 1 \cdot 0 = 4 \)
- First row, second column: \( 2 \cdot 1 + 1 \cdot 3 = 5 \)
- Second row, first column: \( 0 \cdot 2 + 3 \cdot 0 = 0 \)
- Second row, second column: \( 0 \cdot 1 + 3 \cdot 3 = 9 \)
Thus,
\[
X^2 = \begin{pmatrix} 4 & 5 \\ 0 & 9 \end{pmatrix}
\]
### Step 2: Calculate \( X^3 \)
Next, we calculate \( X^3 = X^2 \cdot X \):
\[
X^3 = \begin{pmatrix} 4 & 5 \\ 0 & 9 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}
\]
Calculating the elements:
- First row, first column: \( 4 \cdot 2 + 5 \cdot 0 = 8 \)
- First row, second column: \( 4 \cdot 1 + 5 \cdot 3 = 19 \)
- Second row, first column: \( 0 \cdot 2 + 9 \cdot 0 = 0 \)
- Second row, second column: \( 0 \cdot 1 + 9 \cdot 3 = 27 \)
Thus,
\[
X^3 = \begin{pmatrix} 8 & 19 \\ 0 & 27 \end{pmatrix}
\]
### Step 3: Calculate \( X^6 \)
Now, we calculate \( X^6 = X^3 \cdot X^3 \):
\[
X^6 = \begin{pmatrix} 8 & 19 \\ 0 & 27 \end{pmatrix} \cdot \begin{pmatrix} 8 & 19 \\ 0 & 27 \end{pmatrix}
\]
Calculating the elements:
- First row, first column: \( 8 \cdot 8 + 19 \cdot 0 = 64 \)
- First row, second column: \( 8 \cdot 19 + 19 \cdot 27 = 152 + 513 = 665 \)
- Second row, first column: \( 0 \cdot 8 + 27 \cdot 0 = 0 \)
- Second row, second column: \( 0 \cdot 19 + 27 \cdot 27 = 729 \)
Thus,
\[
X^6 = \begin{pmatrix} 64 & 665 \\ 0 & 729 \end{pmatrix}
\]
### Step 4: Identify \( a, b, c, d \)
From \( X^6 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), we have:
- \( a = 64 \)
- \( b = 665 \)
- \( c = 0 \)
- \( d = 729 \)
### Step 5: Calculate \( a + b + 2020c + d \)
Now we compute:
\[
a + b + 2020c + d = 64 + 665 + 2020 \cdot 0 + 729 = 64 + 665 + 0 + 729 = 1458
\]
### Step 6: Find the number of divisors of 1458
To find the number of divisors of 1458, we first factor it:
\[
1458 = 2 \times 729 = 2 \times 3^6
\]
The formula for the number of divisors based on the prime factorization \( p_1^{e_1} \times p_2^{e_2} \cdots \times p_n^{e_n} \) is:
\[
(e_1 + 1)(e_2 + 1) \cdots (e_n + 1)
\]
For \( 1458 = 2^1 \times 3^6 \):
- For \( 2^1 \): \( 1 + 1 = 2 \)
- For \( 3^6 \): \( 6 + 1 = 7 \)
Thus, the number of divisors is:
\[
2 \times 7 = 14
\]
### Final Answer
The number of divisors of \( a + b + 2020c + d \) is \( \boxed{14} \).
