Memorylessness of the Geometric Distribution

Let $X \sim \text{Geometric}(p)$ count the number of independent Bernoulli($p$) trials until the first success (so $P(X = k) = (1-p)^{k-1} p$ for $k = 1, 2, \ldots$).

(a) Derive a closed-form expression for $P(X > n)$.

(b) Prove the memorylessness property: for all positive integers $m, n$, $$P(X > m + n \mid X > m) = P(X > n).$$

(c) Is there any other discrete distribution on $\{1, 2, 3, \ldots\}$ that is memoryless? Justify briefly.