The smallest positive integer that does not divide $$1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$$ is:

The product $$1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$$ equals $$9! = 362{,}880$$.

To find the smallest positive integer that does not divide $$9!$$ we begin with natural numbers in increasing order.

Every integer from $$1$$ to $$9$$ itself appears as a factor in $$9!$$, so each of them divides $$9!$$.

Next is $$10$$. Since $$10 = 2 \times 5$$ and both $$2$$ and $$5$$ are among the factors of $$9!$$, $$10$$ also divides $$9!$$.

Consider $$11$$. The prime factor $$11$$ is absent from the prime-factorization of $$9!$$ (which contains only the primes $$2, 3, 5,$$ and $$7$$). Therefore $$11$$ does not divide $$9!$$.

Hence, the smallest positive integer that fails to divide the given product is $$11$$.

Answer: 11