Review question

# Can we find integers that satisfy $a^3 + 3b^3 = 9c^3$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7096

## Question

1. 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 $a = b = c = 0$.

2. 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 $p = q = r = 0$.