State the Chinese Remainder Theorem.

Find all solutions to the simultaneous congruences

$$\begin{aligned} &x \equiv 2 \quad(\bmod 3) \\ &x \equiv 3(\bmod 5) \\ &x \equiv 5(\bmod 7) \end{aligned}$$

A positive integer is said to be square-free if it is the product of distinct primes. Show that there are 100 consecutive numbers that are not square-free.