1. Convert the following fractions to decimals:
(i) $\frac{371}{6250}$,
(ii) $\frac{190}{37}$,
(iii) $\frac{13}{28}$,
(iv) $\frac{a}{875}, a=4,29,139,361$.
1. (i) Solution: Since $6250=2 \times 5^{5}$, $\frac{371}{6250}$ is a finite decimal. After calculation, we get
$$\frac{371}{6250}=0.05936$$
(ii) Solution: $\frac{190}{37}=5+\frac{5}{37}$. Since $(10,37)=1$, $\frac{5}{37}$ is a pure repeating decimal. Also, $p(37)=36$, and $10^{2} \neq 1(\bmod 37), 10^{3} \equiv 1(\bmod 37)$, so the length of the repeating cycle is 3. After calculation, we get
$$\frac{190}{37}=5 . \dot{1} 3 \dot{5}$$
(iii) Solution: Since $28=2^{2} \times 7$, $\frac{13}{28}$ is a mixed repeating decimal. Also, $\varphi(7)=6$, and $10^{2} \equiv 1(\bmod 7), 10^{3} \equiv 1(\bmod 7), 10^{6} \equiv 1(\bmod 7)$, so the length of the repeating cycle is 6. After calculation, we get
$$\frac{13}{28}=0.46 \dot{4} 2857 \dot{1}$$
(iv) Solution: Since $875=5^{3} \times 7$, $\frac{a}{875}$ is a mixed repeating decimal. From the calculation in (iii), we know the length of the repeating cycle is 6. After calculation, we get
$$\begin{array}{l}
\frac{4}{875}=0.004 \dot{5} 7142 \dot{8} \\
\frac{29}{875}=0.033 \dot{1} 4285 \dot{7} \\
\frac{139}{875}=0.158 \dot{8} 5714 \dot{2} \\
\frac{361}{875}=0.41257142 \dot{8}
\end{array}$$