I have a set of small prime numbers $S = \{2,3,5,7,11,13,17,19,23,29\}$.
By multiplying those I can form other numbers, by assigning to each element of the set an exponent of $0$ or $1$, so that I can form $2^{10}$ different numbers in total--because there are ten elements in the set. So let's say I call the set of all such numbers $T$.
I also have a (large) prime, around $10^6$, which I call $p$.
Let's call the set of numbers in $T$, reduced $\bmod p$, $T_p$. Are all elements of $T_p$ different, as is the case for $T$? If yes, how would one prove that? If not, please give an example.
This is not an exercise, so I have no reference. The answer might be positive or negative, but I need some input to find out. :) (Also, if anyone can word the title a bit better that I have, please do provide suggestions)
From SAGE: sage: factor$(2*3*5*7*11*17*19*23*29 -13)$
$67 * 7427891$
So the prime number above exceeding 7 million divides the difference between 13 and the product of the nine prime numbers < 30 after excluding 13.