NMC 1993 Problem 4

Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$.

a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \leq k \leq 1992$, but $T(1993 \cdot N)$ is odd. b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.