Answer
See below.
Work Step by Step
1. Use the method of contradiction, suppose the opposite is true: $a$ mod $3$=2, we can write $a=3k+2$
2. case1: $k$ is even, $k=2n$ and $a=6n+2$, we have $a$ mod 6=2 which contradicts the condition $a$ mod 6=3;
3. case2: $k$ is odd, $k=2m+1$ and $a=6m+5$, we have $a$ mod 6=5 which again contradicts the condition $a$ mod 6=3;
4. Thus we can conclude that $a$ mod $3\ne2$ under the given conditions.
