We say that a positive three-digit integer is three stars if it is the result of the product of three distinct prime numbers. For example, $286=2 \cdot 11 \cdot 13$ is a three-star number, but $30=2 \cdot 3 \cdot 5$ and $275=5 \cdot 5 \cdot 11$ are not three-star numbers, as the first has only two digits and the second is not the product of three distinct primes.

(a) What is the smallest three-star number?

(b) Show that every three-star number is divisible by 2, by 3, or by 5.
Solution

(a) We can factor the smallest three-digit numbers $100=2 \cdot 2 \cdot 5 \cdot 5, 101=101$ (it is prime) and $102=2 \cdot 3 \cdot 17$. Since the first two are not three-star numbers, we can conclude that 102 is the smallest three-star number.

(b) If we take a number that is the product of three distinct primes and does not have the factors 2, 3, and 5, it will then be at least $7 \cdot 11 \cdot 13=1001$. Therefore, it will have more than three digits and will not be a three-star number.

Thus, we conclude that every three-star number is divisible by 2, 3, or 5.