Question

prove that an integer is divisible by 3 if and only if sum of its digits is divisible by 3

Answer 1

Let =01…

be the decimal representation of an integer . Thus =+−110+−2102+⋯+010 where 0≤<10 The key observation is that 10≡1(mod3), i.e., [10]=[1]. Hence [10]=[10]=[1], i.e., 10≡1(mod3). The assertion is an immediate consequence of this congruence.