let p and q be two distinct odd prime numbers, n=pq and let d=gcd(p-1,q-1). Then show that x^(ϕ(n)/d) ≡1 (mod n)
I know that ϕ(n)=(p-1)(q-1)
how can i show that???.

x≡a1 (mod p)
x≡a2 (mod q)
we can write x^(p-1(q-1)/d ≡a1 ^ d((p-1)(q-1))/d ≡a1^(p-1)(q-1) ≡1 (mod p)

the same thing for mod q

so x^(ϕ(n)/d) ≡1 (mod pq)≡1 (mod n)
I hope this is correct.

2. We have

$x^{p-1}\equiv1\pmod p$

and

$x^{q-1}\equiv1\pmod q$

Hence $x^{\mathrm{lcm}(p-1,q-1)}\equiv1\pmod n$ (this is because the multiplicative group $\mathbb Z_n^\times$ is isomorphic to $\mathbb Z_p^\times\times\mathbb Z_q^\times).$ The result follows from the fact that

$\mathrm{lcm}(p-1,q-1)=\frac{(p-1)(q-1)}{\gcd(p-1,q-1)}=\frac{\varphi(n)}{d}$

NB: The result also holds if one of $p$ and $q$ is equal to $2.$