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 \(3\) ?

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