Get $n_1,n_2,...,n_r$ positive integers such that $\gcd(n_i,n_j)=1$ always that $i\neq j$ and get $a,b$ two arbitrary integers..
Show that $$a\equiv b \pmod{n_1n_2...n_r}$$ if and only if $$a \equiv b \pmod{n_i}$$ for all $i = 1, 2,. . . , r$
Get $n_1,n_2,...,n_r$ positive integers such that $\gcd(n_i,n_j)=1$ always that $i\neq j$ and get $a,b$ two arbitrary integers..
Show that $$a\equiv b \pmod{n_1n_2...n_r}$$ if and only if $$a \equiv b \pmod{n_i}$$ for all $i = 1, 2,. . . , r$
if $a\equiv b \mod n_1\dots n_r$, then $a=b+k\cdot n_1\dots n_r$, and so $a\equiv b\mod n_i,\quad\forall i$.
If $a\equiv b\mod n_i,\quad\forall i$, then $a=b+k_1n_1=\dots=k_rn_r$. However as $\gcd(n_i,n_j)=1$ for $i\ne j$, each $k_i$ must be divisible by all the other $n_j$, and so $a\equiv b \mod n_1\dots n_r$.