14.1.1 ${ }^{\star}$ Find all positive integers $a$ that satisfy $10 \mid a^{10}+1$.

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Notice that, for any $a \in \mathbf{N}$. There exists $r \in\{0, \pm 1, \pm 2, \pm 3, \pm 4,5\}$, such that $a^{10}+$ $1=r^{10}+1(\bmod 10)$. If and only if $r= \pm 3$, we have $10 \mid r^{10}+1$. Therefore, the positive integer $a=$ $10 k-3$ or $10 k-7, k$ is a positive integer.