Let $N$ be an integer such that we can represent $N$ by its digits as $N=a\_na\_{n-1}\cdots a\_1a\_0$. We want to prove that if $a\_0,a\_1=6$ or if $a\_1=0$ and $a\_0=6$ then $N\equiv 2 \text{ mod } 4$. 

I'm unsure how to proceed here.

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I also want to prove that $N$ is a perfect square only if $N\equiv 0 \text{ mod} 4$ or $N\equiv 1 \text{ mod} 4$. 

**My attempt**
 Through trial with squares $1,2,3$ I notice a cyclic pattern $1,0,1,0...$ as the remainder. I proceeded by induction assuming that $k^2 \equiv 0 \text{ mod} 4$. thus
 $$k^2+1 \equiv 1 \text{ mod} 4$$
But I'm unsure how to get that $2k\equiv 0 \text{ mod } 4$ to complete this. Is the exponentiation identity $a\equiv b \text{ mod } c$ implies that $a^2\equiv b^2 \text{ mod } c$ an iff statement? If so my induction hypothesis gives that $k \equiv 0 \text{ mod } 4$ and the mulitplication rule gives $2k$ congruent to $0$, where I can then apply the addition rule.