3. Suppose that $a, b$ and $c$ are positive integers with $2^{a} 3^{b} 5^{c}=36000$. What is the value of $3 a+4 b+6 c$ ?
3. We note that
$$
36000=36 \times 1000=6^{2} \times 10^{3}=(2 \times 3)^{2} \times(2 \times 5)^{3}=2^{2} \times 3^{2} \times 2^{3} \times 5^{3}=2^{5} \times 3^{2} \times 5^{3}
$$

This is called the prime factorization of 36000 . There are many different ways of getting to this factorization, although the final answer will always be the same.
Since $36000=2^{5} \times 3^{2} \times 5^{3}$ and we want $36000=2^{a} 3^{b} 5^{c}$, then $a=5$ and $b=2$ and $c=3$.
Thus, $3 a+4 b+6 c=3 \times 5+4 \times 2+6 \times 3=15+8+18=41$.
ANSWER: 41