# Meaning of the Corollary 31.29 of the Chinese Remainder Theorem

If $n_1, n_2, ..., n_k$ are pairwise relatively prime and $n = n_1n_2...n_k$ then for all integers $x$ and $a$,

$$x \equiv a \pmod {n_i}$$

for $i=1,2,...,k$ if and only if

$$x \equiv a \pmod n$$

There seems to be some words missing in the theorem/corollary. Should it read:

For all integers $x$ and $a$ that satisfy the equation:

$$x \equiv a \pmod {n_i}$$

we have that it implies that:

$$x \equiv a \pmod n$$

is also true. Furthermore, the converse is true too.

In other words, if: $x \equiv a \pmod {n_i}$ for some integer $a$ and $x$, then it implies that $x \equiv a \pmod n$ and visa versa. i.e. If $x \equiv a \pmod n$ is true then $x \equiv a \pmod {n_i}$.

Is that correct?

• Yes, the meaning is exactly that you mentioned. "If and only if" means that the implication is true in both directions. – Peter Dec 15 '15 at 9:48

It means something like $x \equiv 2 \pmod 5$ and $x \equiv 2 \pmod 7$ if and only if $x \equiv 2 \pmod{35}$