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$L_1$ is regular, having regular expression $(aa)^*(bb)^* +a(aa)^*b(bb)^*$ , either both $m$ and $n$ are even or both are odd then $m+n$ will be even

in case of $L_2$

$L_2=\{a^mb^n\: | m-n=4\}$

$=\{a^mb^n\: | m=n+4\}$

$=\{a^{n+4}b^n\}$

$=\{aaaaa^nb^n\}$
 

$L_2$ is CFL.

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