Question

Show that the square of the positive integer q cannot be of the form 6q+2or 6q+5

Answer 1

We're asked to check if the square of a positive integer $a$ can't be of the form $6q+2$ or $6q+5$ where $q$ is an integer.

Let $a=6m+r$ for some integers $m$ and $r$ where $0\leq r<6.$

This means that $a$ leaves remainder $r$ on division by 6.

$\longrightarrow a\equiv r\pmod{6}$

Squaring we get,

$\longrightarrow a^2\equiv r^2\pmod{6}$

Now let us check if $a^2$ becomes congruent to 2 or 5 modulo 6 in any case. Note that $0\leq r<6.$

Let $r=0.$ Then,

$\longrightarrow a^2\equiv 0^2=\mathbf{0}\pmod{6}$

Let $r=1.$ Then,

$\longrightarrow a^2\equiv 1^2=\mathbf{1}\pmod{6}$

Let $r=2.$ Then,

$\longrightarrow a^2\equiv 2^2=\mathbf{4}\pmod{6}$

Let $r=3.$ Then,

$\longrightarrow a^2\equiv 3^2=9\equiv\mathbf{3}\pmod{6}$

Let $r=4.$ Then,

$\longrightarrow a^2\equiv 4^2=16\equiv\mathbf{4}\pmod{6}$

Let $r=5.$ Then,

$\longrightarrow a^2\equiv 5^2=25\equiv\mathbf{1}\pmod{6}$

None among these cases implied that $a^2$ leaves remainder 2 or 5 on division by 6.

This means $a^2$ cannot be written in the form $6q+2$ or $6q+5.$

Hence Proved!