<s>[INST] A multiple choice question on number theory Pick out the true statement(s):
(a) If $n$ is an odd positive integer, then $8$ divides $n^2 – 1$.
(b) If $n$ and $m$ are odd positive integers, then $n^2+m^2$ is not a perfect square.
(c) For every positive integer $n$, $n^5/5+n^3/3+7n/15$ is an integer.    
(a) is true by induction. But I can not verify the others. Please help me somebody
 [/INST] (a) can also be proved without induction.
$$(2c+1)^2=8\frac{c(c+1)}2+1\equiv1\pmod 8$$
(b)$$(2a+1)^2+(2b+1)^2=4\{a^2+a+b^2+b\}+2\equiv2\pmod 4$$
But any number $m\equiv0,1,2,3\pmod 4\implies m^2\equiv0,1\pmod 4$
(c) Using Fermat's Little Theorem, prime $p\mid(n^p-n) $ for all integer $n$.
So, $$5\mid (n^5-n) \text{  and       }  3\mid(n^3-n)$$
Hence, $$\frac{n^5}5+\frac{n^3}3+\frac{7n}{15}=\frac{n^5-n}5+\frac{n^3-n}3+\left(\frac n3+\frac n5+\frac{7n}{15}\right)=\frac{n^5-n}5+\frac{n^3-n}3+n$$
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