A right-angled triangle has integer sides of length $a,b,c$ where $a<b<c$ and $a,c$ have no common divisor.

Given that the sides of the triangle form an arithmetic sequence, find the value of $b.$

Since $a,b,c$ is an arithmetic sequence, we must have

Rearrange this to find $b$ in terms of $a$ and $c.$

From the previous hint, you should have found $b=\frac{a+c}{2}.$

Substitute this into Pythagoras' Theorem.

Try to manipulate Pythagoras' theorem to show that

Because $a$ and $c$ have no common divisor, that means the fraction $\frac{a}{c}$ is already in simplest terms, so $a=3,c=5.$