Can we find integers that satisfy $a^3 + 3b^3 = 9c^3$?

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Suppose that \(a\), \(b\) and \(c\) are integers that satisfy the equation \[a^3 + 3b^3 = 9c^3.\]

Explain why \(a\) must be divisible by \(3\), and show further that both \(b\) and \(c\) must also be divisible by \(3\). Hence show that the only integer solution is \(a = b = c = 0\).

Suppose that \(p\), \(q\) and \(r\) are integers that satisfy the equation \[p^4 + 2q^4 = 5r^4.\]

By considering the possible final digit of each term, or otherwise, show that \(p\) and \(q\) are divisible by \(5\). Hence show that the only integer solution is \(p = q = r = 0\).