Three got mad!

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
1 year, 3 months ago

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