Theorem: An integer is a multiple of nine if and only if the sum of its digits is a multiple of nine.
Proof: Express the integer in terms of its digits: \(d_0+10d_1+100d_2+\dots+10^nd_n\). This may be grouped into the sum of its digits \(d_0+d_1+\dots+d_n\) plus the sum \((10-1)d_1+(100-1)d_2+\dots+(10^n-1)d_n\). Since each \(10^i-1\) is divisible by nine…
