<s>[INST] Prove or disprove : If $a\equiv b$ mod $m$, when $a,b,m\in \mathbb{Z}$ , then $a^3\equiv b^3$ mod $m^2$. Prove or disprove : If $a\equiv b$ mod $m$, when $a,b,m\in \mathbb{Z}$ , then 
$a^3\equiv b^3$ mod $m^2$
My attempt:
since  $a\equiv b$ mod $m$ then $a=b+mk$ for some $k$
Now $a^3=m^3k^3+b^3+3m^2k^2b+3mkb^2$
$a^3-b^3=m^3k^3+3m^2k^2b+3mkb^2$ 
from here I stuck 
 [/INST] $$1\equiv 5\pmod 4$$
but $$5^3\equiv 13\not\equiv 1\pmod{16}.$$
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