For any a ∈ N, prove that a 3 − a is divisible by 3. [Hint: Split into three cases: a = 3 n , a = 3 n + 1, and a = 3 n + 2 for some n (and explain why these cover all possibilities).] [Another hint: (...

For any
a
∈ N, prove that
a
³
−
a
is divisible by 3. [Hint: Split into three cases:
a
= 3n,
a
= 3n
+ 1, and
a
= 3n
+ 2 for some
n
(and explain why these cover all possibilities).] [Another hint: (x
+
y)³
=
x
³
+ 3x
²
y
+ 3xy
²
+
y
³
for any
x
and
y.]