1. (Modular arithmetic.) (a) Take any integer N. Show that N is a multiple of three if and only if the sum of the digits in N is a multiple of 3. (For example: 729 is a multiple of 3 because 7 + 2 + 9 = 18 is a multiple of 3.)

(b) We say that a number N is a rainbow number if every digit from {0, 1, . . . 9} shows up exactly once in N; for example, 1234567890 and 1029384756 are both rainbow numbers. Take any two integer rainbow numbers N, M. Prove that N M 0 mod 9.