smc

Given that $p$ is odd, $p-1$ must be even, therefore $\frac{1}{2}(p-1)$ must be an integer, which will be denoted as n. $$(p-1{)}^n-1$$ By sum and difference of powers $$=((p-1)-1)((p-1{)}^{n-1}+\cdots +1^{n-1})$$ $$=(p-2)((p-1{)}^{n-1}+\cdots +1^{n-1})$$ $p-2$ divide $(p-1{)}^{\frac{1}{2}(p-1)}$$\boxed{(p-1{)}^{\frac{1}{2}(p-1)}-1\text{is divisible by}p-2}$.