Show that the square of any integer is of the form $3m$ or $3m+1$ but not of the form $3m+2$ .

Step 1: Identify the possible remainders when an integer is divided by 3 When an integer is divided by 3, there can be three possible remainders: 0, 1, or 2. Step 2: Analyze the case where the remainder is 0 Let's consider an integer $n$ where its remainder when divided by 3 is 0. Then, we can write $n$ as $n=3k$ for some integer $k$. When we square $n$, we have: $\begin{array}{rl}{n}^2& =(3k{)}^2\\ & =9k^2\\ & =3(3k^2)\end{array}$ Since $3k^2$ is an integer, the square of $n$ is of the form $3m$ where $m=3k^2$. Step 3: Analyze the case where the remainder is 1 Let's consider an integer $n$ where its remainder when divided by 3 is 1. Then, we can write $n$ as $n=3k+1$ for some integer $k$. When we square $n$, we have: $\begin{array}{rl}{n}^2& =(3k+1{)}^2\\ & =9k^2+6k+1\\ & =3(3k^2+2k)+1\end{array}$ Since $3k^2+2k$ is an integer, the square of $n$ is of the form $3m+1$ where $m=3k^2+2k$. Step 4: Analyze the case where the remainder is 2 Let's consider an integer $n$ where its remainder when divided by 3 is 2. Then, we can write $n$ as $n=3k+2$ for some integer $k$. When we square $n$, we have: $\begin{array}{rl}{n}^2& =(3k+2{)}^2\\ & =9k^2+12k+4\\ & =3(3k^2+4k)+1\end{array}$ Since $3k^2+4k$ is an integer, the square of $n$ is of the form $3m+1$ where $m=3k^2+4k$. Step 5: Conclusion We analyzed all possible cases of integer remainders when divided by 3. In each case, we found that the square is either of the form $3m$ or $3m+1$. We never had a square in the form of $3m+2$. Therefore, we can conclude that the square of any integer is of the form $3m$ or $3m+1$, but not of the form $3m+2$.