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Solution: The sum of the divisors of $ 2^3 \cdot 3^2 $ is given by:
Thus, the sum of all positive divisors is $ \boxed{195} $.
Question: What is the remainder when $ 7^5 + 7^6 $ is divided by 6?
Solution: First, note that $ 7 \equiv 1 \mod 6 $, so $ 7^k \equiv 1^k = 1 \mod 6 $ for any positive integer $ k $.
7^5 + 7^6 \equiv 1 + 1 = 2 \mod 6
Thus, the remainder is $ \boxed{2} $.
Question: What is the largest prime factor of $ 2^4 \cdot 3^3 \cdot 5^2 $?
Solution: The prime factorization is already given: $ 2^4 \cdot 3^3 \cdot 5^2 $. The largest prime factor is clearly $ \boxed{5} $.
Question: What is the smallest positive integer divisible by both 12 and 18?
Solution: To find the least common multiple of 12 and 18, factor both: