answer

$$4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.$$$2^{2a} \cdot 5^b \cdot 2^c \cdot 3^c = 2^{24} \cdot 3^{18} \cdot 5^{10} \cdot 2^{10}.$ $2^{2a+c-34} \cdot 3^{c-18} \cdot 5^{b-10}=1$ The only possible solution is when all exponents are $0$ $c=18$ $2a+c = 34; $ $a = 8$ $b=10$ $a+b+c=18+8+10=36$