If each prime divides $a$, then the product of primes divides $a$?

Question

If $p_1, p_2,...,p_k$ are distinct primes and each $p_i \mid a$ , $i=1, 2,...,k$ does this implies that $p_1 \cdot p_2 \cdots p_k \mid a$?

Answer 1

Yes, unless the primes are not all distinct. say for example $p_1 = 3, p_2 = 3, p_3 = 3, a = 3$, the product 27 for certain does not divide three

But if they are distinct, then the product would divide since every natural number except one has only one unique prime factorisation.