Source: 2017 Canadian Open Math Challenge, Problem A4
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Three positive integers $a$, $b$, $c$ satisfy

$$4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.$$

Determine the sum of $a + b + c$.

1. Start with the given equation:
   \[
   4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}
   \]

2. Express each term as a product of prime factors:
   \[
   4 = 2^2, \quad 5 = 5, \quad 6 = 2 \cdot 3
   \]
   \[
   8 = 2^3, \quad 9 = 3^2, \quad 10 = 2 \cdot 5
   \]

3. Substitute these into the equation:
   \[
   (2^2)^a \cdot 5^b \cdot (2 \cdot 3)^c = (2^3)^8 \cdot (3^2)^9 \cdot (2 \cdot 5)^{10}
   \]

4. Simplify the exponents:
   \[
   2^{2a} \cdot 5^b \cdot 2^c \cdot 3^c = 2^{24} \cdot 3^{18} \cdot 2^{10} \cdot 5^{10}
   \]

5. Combine like terms on the right-hand side:
   \[
   2^{2a+c} \cdot 3^c \cdot 5^b = 2^{34} \cdot 3^{18} \cdot 5^{10}
   \]

6. Equate the exponents of the corresponding prime factors:
   - For the base 2:
     \[
     2a + c = 34
     \]
   - For the base 3:
     \[
     c = 18
     \]
   - For the base 5:
     \[
     b = 10
     \]

7. Solve for \(a\) using \(c = 18\):
   \[
   2a + 18 = 34
   \]
   \[
   2a = 16
   \]
   \[
   a = 8
   \]

8. Sum the values of \(a\), \(b\), and \(c\):
   \[
   a + b + c = 8 + 10 + 18 = 36
   \]

The final answer is \(\boxed{36}\)