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 $275=5 \cdot 5 \cdot 13$ 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 each three-star number has some common divisor with 30 greater than 1.
Solution

(a) The first two three-digit numbers are $100=2 \cdot 2 \cdot 5 \cdot 5$ and $101=101$ (which is prime). When testing 102, we have $102=2 \cdot 3 \cdot 17$, which is the smallest three-star number.

(b) It suffices to show that every three-star number has at least one of the prime factors in the set $\{2,3,5\}$. If we take a three-star number that does not have at least one of these factors, it will be at least $7 \cdot 11 \cdot 13=1001$ and, consequently, will have more than three digits and will not be a three-star number.