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For every real positive integer $$n = \overline{a_ka_{k-1}a_{k-2}...a_0}$$ can be written as $$n = a_0 \times 10^0 + a_1 \times 10^1 + ... + a_k \times 10^k$$. For example, $$576 = 5 \times 10^2 + 7 \times 10^1 + 6 \times 10^0$$.

For every real positive integer $$A$$, let $$S(A)$$ denotes the sum of the digits of $$A$$. For example, $$S(576) = 5 + 7 + 6 = 18$$.

Question 1. For real positive integer $$N = \overline{a_ka_{k-1}a_{k-2}...a_0}$$, prove that $$S(N) - N$$ is divisible by $$3$$.

Question 2. From Q1, please explain why is $$N$$ divisible by $$3$$ if and only if $$S(N)$$ is divisible by $$3$$

Question 3. Let $$m$$ and $$n$$ be two positive real integer. And let $$x$$ be a number that is obtained by concatenating $$m$$ and $$n$$. For example, if $$m = 23$$ and $$n = 546$$, then $$x = 23546$$. Prove that $$S(x) - S(m) - S(n)$$ is divisible by $$3$$. And also, prove that $$x = m + n \space (\text{mod} \space 3)$$.

Question 4. A sequence $$b_1 , b_2 , b_3, ...$$ defined as written below.

$$b_1 = 1; \quad b_2 = 12; \quad b_3 = 123; \quad b_4 = 1234; b_5 = 12345; \space b_6 = 123456 \space \text{etc}$$.

For every positive integer $$k$$, prove that $$b_{3k + 1}$$ is not divisible by $$3$$.

Question 5. Based on the sequence from Q4, among $$b_1 , b_2 , b_3 , ... , b_{2017}$$, how many number are there that are divisible by $$3$$ ?

Note by Fidel Simanjuntak
3 months, 3 weeks ago