I was reading A. L. Whiteman - A family of difference sets. On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. At the last paragraph can someone explain why if $p-1=df$ and $q-1=ef'$ and $(f,f')=1$, if $ff'$ is odd, then $-1 \equiv g^{(d/2)} \pmod v$ and if $ff'$ is even, there does not exist any $s$, $s=0,\dotsc, d-1$, where $d$ is $\operatorname{lcm}(p-1,q-1)$ such that $-1 \equiv g^{s}\pmod v$?

I was reading A. L. Whiteman - A family of difference sets. On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. At the last paragraph, can someone explain why if $p-1=df$ and $q-1=ef'$ and $(f,f')=1$, if $ff'$ is odd, then $-1 \equiv g^{(d/2)} \pmod v$, and if $ff'$ is even, there does not exist any $s$, $s=0,\dotsc, d-1$, where $d$ is $\operatorname{lcm}(p-1,q-1)$ such that $-1 \equiv g^{s}\pmod v$?

I was reading A. L. Whiteman - A family of difference sets. On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. At the last paragraph can someone explain if $p-1=df$ and $q-1=ef'$ and $(f,f')=1$, if $ff'$ is odd, then $-1 \equiv g^{(d/2)} \pmod v$ and if $ff'$ is even, there does not exist any $s$, $s=0,\dotsc, d-1$, where $d$ is $\operatorname{lcm}(p-1,q-1)$ such that $-1 \equiv g^{s}\pmod v$.

I was reading A. L. Whiteman - A family of difference sets. On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. At the last paragraph can someone explain why if $p-1=df$ and $q-1=ef'$ and $(f,f')=1$, if $ff'$ is odd, then $-1 \equiv g^{(d/2)} \pmod v$ and if $ff'$ is even, there does not exist any $s$, $s=0,\dotsc, d-1$, where $d$ is $\operatorname{lcm}(p-1,q-1)$ such that $-1 \equiv g^{s}\pmod v$?