  Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod
_{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the
properties of the modular equations of the second and the third order under the
Atkin $U$-operator, we determine the generating functions of
$p_{8k}(2^{2\alpha} n +\frac{k(2^{2\alpha}-1)}{3})$ $(1\leq k\leq 3)$ and
$p_{3k} (3^{2\beta}n+\frac{k(3^{2\beta}-1)}{8})$ $(1\leq k\leq 8)$ in terms of
some linear recurring sequences. Combining with a result of Engstrom about the
periodicity of linear recurring sequences modulo $m$, we obtain infinite
families of congruences for $p_k(n)$ modulo any $m\geq2$, where $1\leq k\leq
24$ and $3|k$ or $8|k$. Based on these congruences for $p_k(n)$, infinite
families of congruences for many partition functions such as the overpartition
function, $t$-core partition functions and $\ell$-regular partition functions
are easily obtained.
