A student claims that 180 is a term of the AP 4, 13, 22, 31, ... Is the claim correct?

$a=4,d=9$. If $180=4+(n-1)9$ then $176=9(n-1)$, so $n-1=19.6$ and $n=20.6$. A term number must be a positive whole number, so 180 is not a term (indeed $a_{20}=175$ and $a_{21}=184$). The last option gives the wrong reason: no term of this AP is a multiple of 9 either, since every term leaves remainder 4.