Q:

Prove the following divisibility statements without use of induction

(a) $5$ $|$ $3^{3n+1}+2^{n+1}$
(b) $21$ $|$ $4^{n+1} + 5^{2n-1}$
(c) $24$ $|$ $2 \cdot7^n + 3 \cdot5^n - 5$
These are trivial by using induction.  But I have tried to prove it by binomial theorem and remainder theorem, but couldn't succeed.  Please help to prove it. 

A:

Hints with modular arithmetic:
$$(a)\;\;3^3=2\pmod 5\implies \left(3^{3}\right)^n\cdot 3=2^n\cdot 3\implies 3^{3n+1}+2^{n+1}=2^n(3+2)\pmod 5$$
$$(b)\;\;5^2=4\pmod {21}\implies 4^{n+1}+5^{2n-1}=4^n(4+5^{-1})\pmod{21}\;,\;$$
$$\text{but}\;5^{-1}=-4\pmod{21}\ldots$$
Now you try something similar as the above for the third one

