  We prove an asymptotic formula for the number of integers $\leq x$ which can
be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with
each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j,
q)=1$ $(q \geq 3, 1\leq j\leq k)$. For any $A>0$, our result is uniform for
$2\leq k\leq A\log\log x$. Moreover, we show that, there are large biases
toward certain arithmetic progressions $(a_1 \bmod q, \cdots, a_k \bmod q)$,
and such biases have connections with Mertens' theorem and the least prime in
arithmetic progressions.
