To solve the problem, we need to find the smallest number with exactly four different prime factors.

Step-by-step Solution:

1. The smallest prime numbers are $2, 3, 5, \text{and } 7$.
2. The smallest number with exactly four different prime factors will be the product of these four smallest primes: $2 \times 3 \times 5 \times 7 = 210$

Answer:

The smallest number with exactly four different prime factors is 210.

Would you like a detailed breakdown or have any questions about this solution? Here are some related questions to explore:

1. How can we verify the number of prime factors of a given number?
2. What is the smallest number with five different prime factors?
3. How do prime numbers relate to the factorization of composite numbers?
4. What is the largest prime number under 50?
5. How can we calculate the least common multiple (LCM) using prime factors?

Tip: Remember, the smallest numbers with specific numbers of distinct prime factors are always products of the smallest