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 years, 11 months ago

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