There are real numbers $a, b, c, $ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i, $ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$
1. **Identify the roots and use Vieta's formulas:**
   - For the polynomial \(x^3 + ax + b\), we know that \(-20\) is a root. Let the other roots be \(m + \sqrt{n}i\) and \(m - \sqrt{n}i\) (since complex roots come in conjugate pairs).
   - The sum of the roots of \(x^3 + ax + b = 0\) is zero (since the coefficient of \(x^2\) is zero). Therefore:
     \[
     -20 + (m + \sqrt{n}i) + (m - \sqrt{n}i) = 0
     \]
     Simplifying, we get:
     \[
     -20 + 2m = 0 \implies 2m = 20 \implies m = 10
     \]

2. **Use the second polynomial and Vieta's formulas:**
   - For the polynomial \(x^3 + cx^2 + d = 0\), we know that \(-21\) is a root. The other roots are \(10 + \sqrt{n}i\) and \(10 - \sqrt{n}i\).
   - The sum of the roots of \(x^3 + cx^2 + d = 0\) is \(-c\). Therefore:
     \[
     -21 + (10 + \sqrt{n}i) + (10 - \sqrt{n}i) = -c
     \]
     Simplifying, we get:
     \[
     -21 + 10 + 10 = -c \implies -1 = -c \implies c = 1
     \]

3. **Calculate the product of the roots:**
   - The product of the roots of \(x^3 + cx^2 + d = 0\) is \(-d\). Therefore:
     \[
     (-21) \cdot (10 + \sqrt{n}i) \cdot (10 - \sqrt{n}i) = -d
     \]
     Simplifying, we get:
     \[
     (-21) \cdot (10^2 - (\sqrt{n}i)^2) = -d
     \]
     \[
     (-21) \cdot (100 - (-n)) = -d
     \]
     \[
     (-21) \cdot (100 + n) = -d
     \]
     \[
     21 \cdot (100 + n) = d
     \]

4. **Use the fact that the sum of the products of the roots taken two at a time is zero:**
   - The sum of the products of the roots taken two at a time for \(x^3 + cx^2 + d = 0\) is zero. Therefore:
     \[
     (-21)(10 + \sqrt{n}i) + (-21)(10 - \sqrt{n}i) + (10 + \sqrt{n}i)(10 - \sqrt{n}i) = 0
     \]
     Simplifying, we get:
     \[
     -210 - 21\sqrt{n}i - 210 + 21\sqrt{n}i + 100 - n = 0
     \]
     \[
     -420 + 100 + n = 0
     \]
     \[
     -320 + n = 0 \implies n = 320
     \]

5. **Calculate \(m + n\):**
   \[
   m + n = 10 + 320 = 330
   \]

The final answer is \(\boxed{330}\).