What is the smallest positive integer with exactly 10 positive integer divisors?
Recall that the number of whole number divisors of a positive integer can be determined by prime factorizing the integer, adding 1 to each of the exponents, and multiplying the results.  If a positive integer has 10 factors, then the set of exponents in the prime factorization is $\{1,4\}$ or $\{9\}$.  For each set of exponents, the smallest positive integer whose prime factorization has the given set of exponents is achieved by assigning the exponents in decreasing order to the primes 2, 3, 5, etc.  The smallest positive integer with an exponent of 9 in the prime factorization is $2^9=512$.  The smallest positive integer whose prime factorization has exponents 1 and 4 is $2^4\cdot 3^1=48$.  Since $48<512$, $\boxed{48}$ is the smallest positive integer with 10 positive integer divisors.
48
Level 3
Number Theory
Problem: What is the smallest positive integer with exactly 10 positive integer divisors?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the smallest positive integer with exactly 10 positive integer divisors, we need to understand how the number of divisors is determined. The number of divisors of a positive integer n, expressed in its prime factorization form as n = p1^e1 * p2^e2 * ... * pk^ek, is given by the formula (e1 + 1)(e2 + 1)...(ek + 1). 

To achieve exactly 10 divisors, we can factor 10 into its possible forms: 
1. 10 = 10 (which corresponds to a prime raised to the 9th power, p^9)
2. 10 = 5 * 2 (which corresponds to a product of two distinct primes, p1^4 * p2^1)

Next, we evaluate each case to find the smallest integer.

For the first case, p^9, the smallest prime is 2. Thus, 2^9 = 512. 

For the second case, p1^4 * p2^1, we can use the smallest two primes, 2 and 3. This gives us:
2^4 * 3^1 = 16 * 3 = 48.

Now we compare the results from both cases. The smallest integer from the first case is 512, while from the second case, it is 48. 

Since 48 is less than 512, we conclude that the smallest positive integer with exactly 10 positive integer divisors is 48.<|END_OF_LATENT|> The final answer is $48$.