The three sides of a right-angled triangle have lengths $a$, $b$ and $c$, where $a$, $b$, $c \in \mathbb{Z}$ - AQA - A-Level Maths Pure - Question 6 - 2019 - Paper 3

Assume that $a$ and $b$ are both odd. We can express them as:

• $a = 2m + 1$,
• $b = 2n + 1$

for some integers $m$ and $n$. According to the Pythagorean theorem, we have:

$c^2 = a^2 + b^2$

Substituting $a$ and $b$ gives:

$c^2 = (2m + 1)^2 + (2n + 1)^2$

Expanding this results in:

$c^2 = (4m^2 + 4m + 1) + (4n^2 + 4n + 1)$

which simplifies to:

$c^2 = 4m^2 + 4m + 4n^2 + 4n + 2$

This can be rewritten as:

$c^2 = 2(2m^2 + 2m + 2n^2 + 2n + 1)$

Since $c^2$ is even, it follows that $c$ must be even (as the square of an odd number is odd). This leads to a contradiction, as we started with the assumption that $a$, $b$, and $c$ are all odd. Thus, it is not possible for all of $a$, $b$, and $c$ to be odd.