Here is another way of finding the least non negative solution of each system of congruence.

Maybe some of you will find this alternative much easier or harder than the method we learn in class.

Find the least non negative solution of each system of congruence below.

(1)Suppose we have a solution such that x=5a+2, $$ a \in \mathbb{Z}$

$5a+2 \equiv 4 \mod{7}$

$5a \equiv 2 \mod{7}$

$a \equiv 6 \mod{7}$

a = 7b +6, $$ b \in \mathbb{Z}$

So we have:

x=5a+2

x=5(7b+6)+2

x=35b+32

$35b+32 \equiv 3 \mod{9}$

$35b \equiv -29 \mod{9}$

$35b\equiv -2 \mod{9}$

$b \equiv 2 \mod{9}$

b= 9c+2, $$ c \in \mathbb{Z}$

Now we have:

x=35b+32

x=35(9c+2)+32

x=315c+102

So all solutions are:

{315c+102: $$ c \in \mathbb{Z}$}

The smallest positive solution is 102.