Tests for divisibility by 19 and 7

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13. What conditions must the numbers of the basic sequence $q_1, q_2, q_3, \dots$ satisfy in order for the numbers to contain only two digits: 0 and 1? What other conditions should be imposed on these numbers $q_1, q_2, \dots$ so that no two ones come in a row?

14. Sequence of numbers

$$q_0 = 1, \quad q_1 = 2, \quad q_2 = 3,$$

$$q_3 = 5, \quad q_4 = 8, \quad q_5 = 13, \dots,$$

where $q_{n+1} = q_n + q_{n-1}$ is called the Fibonacci sequence. Prove that the Fibonacci system of numbers satisfies all the conditions of the previous problem. Find the sum of numbers in this system

a) $\underbrace{100 \dots 00}_{k} + \underbrace{100 \dots 00}_{l}$;

b) $\underbrace{10101 \dots 01}_{2m+1} + 1$;

c) $\underbrace{10101 \dots 01}_{2m+1} + \underbrace{100 \dots 00}_{2m+1}$.

15. Is it possible to divide all natural numbers 1, 2, 3, 4, 5, 6, 7, … into two increasing sequences $a_1, a_2, a_3, a_4, \dots$ and $b_1, b_2, b_3, b_4, \dots$ such that $b_k - a_k = k$ for any $k = 1, 2, 3, \dots$?

After some thought, you will, of course, agree on how to choose such sequences (this can be done in the only way)

$$\begin{array}{cccccccc} a_1 & b_1 & a_2 & a_3 & b_2 & a_4 & b_3 & a_5 & a_6 & b_4 & a_7 \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ a_8 & b_5 & c_9 & b_6 & \dots \\ 12 & 13 & 14 & 15 & \dots \\ \end{array}$$

It is not so easy to satisfy pairs $(a_k, b_k)$ in education. How can you find out, for example, which of the two sequences $-a_k$ or $b_k$ 100 belongs to, what number it corresponds to, what number will be with it (without writing out all the previous $a_k$ and $b_k$)?

It turns out that in order to formulate this pattern, you need to write these sequences not in the decimal, but in the Fibonacci number system. Try to discover and prove it.

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