∆ Definition 1. Numbers $a$ and $b$ are called congruent modulo $m\ne0$, if $a-b⋮ m$. Notation: $a\equiv b\pmod m$.

Problem 1. Show that $a$ and $b$ are congruent modulo $m$ if and only if the remainder from dividing $a$ by $m$ is equal to the remainder from dividing $b$ by $m$. Send a solution

Problem 2. Show that the congruence relation satisfies all the conditions of an equivalence relation Send a solution

Problem 3. Show that for any $a_1$, $a_2$, $b_1$, $b_2$, $c$, $m$ the following is true 1) $a_1\equiv b_1\pmod m,\;a_2\equiv b_2\pmod m \Longrightarrow a_1+a_2\equiv b_1+b_2\pmod m$; 2) $a_1\equiv b_1\pmod m \Longrightarrow ca_1\equiv cb_1\pmod m$; 3) $a_1\equiv b_1\pmod m,\;a_2\equiv b_2\pmod m \Longrightarrow a_1a_2\equiv b_1b_2\pmod m$. Send a solution

Problem 4. Show that if $a\equiv b\pmod m$, then 1) $a^n\equiv b^n\pmod m$ for any non-negative $n$; 2*) for any polynomial $f(x)$ with integer coefficients $f(a)\equiv f(b)\pmod m$. Send a solution

Problem 5. Is it always true that if $a\equiv b\pmod m$ and $a,b\geqslant0$, then $2^a\equiv 2^b\pmod m$? Send a solution

Problem 6. Let $\overline{\mathstrut a_na_{n-1}\ldots a_1a_0}$ be a digit expression of number $x$. Show that 1) $x\equiv a_0+\ldots+a_n\pmod3$, $x\equiv a_0+\ldots+a_n\pmod9$; 2) $x\equiv a_0\pmod2$, $x\equiv a_0\pmod5$; 3) $x\equiv a_0-a_1+\ldots+(-1)^n a_n\pmod{11}$. Send a solution

Problem 7. Show that if $x$ is odd, then $x^2\equiv1\pmod 8$. Send a solution

Problem 8. * Show that the following equations have no non-zero solutions in integer numbers 1) $x^2 + y^2 = 3z^2$; 2) $x^2 + y^2 + z^2 = 4t^2$. Send a solution

Problem 9. * Show that there are infinitely many natural numbers that cannot be expressed as a sum of three 1) squares; 2) cubes of natural numbers. Send a solution

Problem 10. Solve the following congruent relations а) 3x ≡ 1 (mod 7); б) 6x ≡ 5 (mod 9); в) 4x ≡2 (mod 10). Send a solution

Problem 11. Show that relation $ax\equiv b\pmod m$ has a solution if and only if $b⋮GCD(a,m)$. Send a solution

Problem 12. Let $p$ be a prime number and $a\not\equiv0\pmod p$. Show that relation $ax\equiv b\pmod p$ has a solution, moreover, each two solutions are congruent modulo $p$. Send a solution

Problem 13. * (Chinese remainder theorem) Let numbers $a_1$, $a_2$, ..., $a_n$ be pairwise co-prime (i.e.the only positive integer that divides both of them is 1). Then for any $x_1$, $x_2$, ..., $x_n$ there exists $x$ such that \begin{equation*} x\equiv x_i\pmod {a_i}   i=1,\ldots,n, \end{equation*} moreover, any two numbers, which satisfy this property, are congruent modulo $a_1 a_2 \ldots a_n$. Send a solution

Problem 14. Find all solutions of the following system of equations \begin{equation*} \left\{ \begin{aligned} x&\equiv3\pmod5 \\ x&\equiv1\pmod7 \\ x&\equiv4\pmod9. \end{aligned} \right. \end{equation*} Send a solution

Problem 15. Let $p$ be a prime number. Show that 1) $C_p^k⋮ p$ for $0< k< p$; 2) $(a+b)^p\equiv a^p+b^p\pmod p$; 3) (Fermat's little theorem)  $ (a^p-a ) ⋮ p$. Send a solution

Problem 16. * (Wilson's theorem) Let $p$ be a prime number. Show that $(p-1)!\equiv-1\pmod p$. Send a solution

Problem 17. A tessellation of a plane is the tiling of a plane using geometric shapes, called tiles, with no overlaps and no gaps. Which regular polygons can make a tessellation of a plane? Send a solution

Problem 18. Show that there are infinitely many prime numbers of the following form 1) $4n+3$;  2*) $4n+1$;  3**) $an+b$, where $GCD(a,b)=1$. Send a solution

Problem 19. * Find the number of solutions of equation $x^2\equiv1\pmod n$, where 1) $n>2$ is a prime number; 2) $n=2$; 3) $n$ is any natural number. Send a solution