Problem 1. Are the following statements true for all $n>1$? 1) $6 \nmid n^3 + 5n$; 2) $2n^3 + 3n^2 + 7n ⋮ 6$; 3) $n^5 - n ⋮ 30$; 4) $2^{2n} - 1 ⋮ 6$; 5) $11^{6n + 3} + 1 ⋮ 148$? Send a solution

Problem 2. Define 1) the $GCD$; 2) the $LCM$ of numbers $a_1$, $a_2$, ..., $a_n$ ($n > 2$). Send a solution

Problem 3. Show that for any $a$, $b$ and $c$ such that $a\cdot b\cdot c\ne0$, we have 1) $GCM(a,b,c) = GCD(a,НОД(b,c)) = GCD(GCD(a,b),c)$; 2) $LCM(a,b,c) = \frac{|abc|\cdot GCD(a,b,c)}{GCD(a,b)\cdot GCD(b,c)\cdot GCD(a,c)}$. Send a solution

Problem 4. Is there a number such that the remainders when dividing by $2$, $3$, $4$, $5$, $6$ are correspondingly 1) $1$, $2$, $3$, $4$, $5$; 2) $0$, $1$, $2$, $3$, $4$; 3) $0$, $1$, $2$, $3$, $2$? Send a solution

Problem 5. Compute the $GCD$ of the following numbers 1) $923$ and $1207$; 2) $279$ and $-589$; 3) $-693$ and $2475$; 4) $-697$ and $-1377;$ 5) $1517$ and $1591$; 6) $1134$, $2268$ and $1575$. Send a solution

Problem 6. Compute the $LCM$ of the following numbers 1) $16$ and $84$; 2) $819$ and $504$; 3) $30$, $56$ and $72$; 4) $340$, $990$ and $46$; 5) $41$, $85$ and $36$; 6) $2,5,7,9$ and $11$. Send a solution

Problem 7. For $n>0$ compute the following 1) $1\cdot2 + 2\cdot3 + \ldots + (n-1)\cdot{n}$; 2) $\frac{1}{4\cdot5} + \frac{1}{5\cdot6} + \ldots + \frac{1}{(n+3)(n+4)}$; 3*) $1\cdot2\cdot3 + 2\cdot3\cdot4 + \ldots + n\cdot(n+1)\cdot(n+2)$. Send a solution

Problem 8. Show that the following is true 1) $(n+1)\cdot(n+2)\cdot\ldots\cdot(n+n) = 2^n\cdot1\cdot3\cdot5\cdot\ldots\cdot(2n - 1)$; 2) $1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{1}{2n-1} - \frac{1}{2n} = \frac{1}{n + 1} + \frac{1}{n + 2} + \ldots + \frac{1}{2n}$; 3) $(1 - \frac{1}{4})\cdot(1 - \frac{1}{9})\cdot \ldots \cdot(1 - \frac{1}{(n+1)^2}) = \frac{n + 2}{2n + 2}$. Send a solution

Problem 9. Solve the following equations in integer numbers 1) $7x + 5y = 1$;   2) $27x - 24y = 1$;   3) $12x - 33y = 9$; 4) $-56x + 91y = 21$;   5) $344x - 215y = 86$;   6) $3x + 5y +7z = 1$. Send a solution

Problem 10. Is it true that for any $n$ the numbers $10n+7$ and $10n+5$ are co-prime? Send a solution

Problem 11. Find numbers $a$ and $b$ such that $ax + by = 1$, where 1) $x=7 , y=9$;   2) $x=17 , y=19$;   3) $x=27 , y=29$;   4) $x=37 , y=39$;    5) $x=47 , y=49$.   Send a solution

Problem 12. (the Fibonacci numbers) Let $u(n)$ be a sequence of numbers such that $u(0)=0, u(1)=1, u(n)=u(n-1) + u(n-2)$. 1) Show that $u(1) + \ldots + u(n) = u(n+2) - 1$. 2) Show that $(u(1))^2 + \ldots + (u(n))^2 = u(n)\cdot{u(n+1)}$. 3) (Binet's formula) What do you think is the relation between numbers $u(n)$ and $\delta(n)= \frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^n-\frac{1}{\sqrt{5}} (\frac{1-\sqrt{5}}{2})^n$? Send a solution