number theory pls help!!

We know that

gcd(a,b) \cdot lcm(a,b) = ab

for all positive integers a and b.

Plugging in the given information, we get

18 \cdot lcm(n,24) = 180 \cdot n

Solving for n, we get

lcm(n,24) = 10n

Since 24 = 2^3 * 3, the prime factorization of n must include at least 2^3 and 3.

The smallest such positive integer n is 2^3 * 3 = 24.

Checking, we can see that gcd(24, 180) = 18 and lcm(24, 24) = 504.

Therefore, the only positive integer n satisfying the given conditions is 24​.