Agreement: all numbers on this page are integer numbers.
∆ Definition 1. An integer number $a$ is divisible by a non-zero integer number $b$ if there is an integer number $k$ such that $a=kb$. It is said that $b$ divides $a$. $b$ is called a divisor of $a$. Notation: $a⋮ b$ or $b \mid a$.
Problem 1. Show that for any $a$ the following is true: 1) if $a\ne 0$, then $a⋮ a$; 2) $a⋮ 1$; 3) if $a\ne0$, then $0⋮ a$. Send a solution
Problem 2. Show that for any $a$, $b$, $c$, $x$, $y$ the following is true: 1) if $b \mid a$ and $c\mid b$, then $c\mid a$; 2) if $a⋮ b$ and $a\ne0$, then $|a|\geqslant|b|$; 3) if $c\ne0$, then $a⋮ b\Longleftrightarrow ac⋮ bc$; 4) if $a⋮ b$ and $c⋮ b$, then $(a\pm c)⋮ b$; 5) if $a⋮ b$ and $c⋮ b$, then $ax+cy ⋮ b$; 6) if $a⋮ b$ and $b⋮ a$, then $a=b$ или $a=-b$; 7) if $a⋮ b$, then $ac⋮ b$; 8) if $a⋮ b$ and $b \nmid c$, then $b \nmid (a+c)$; 9) if $ab=cd$ and $a⋮ c$, then $d⋮ b$. Send a solution
Problem 3. Are the following statements true for any $a$, $b$, $c$, $d$? 1) If $b\mid a$ and $c\not\mid b$, then $c\mid a$; 2) if $b\mid a$ and $c\mid a$, then $bc\mid a$; 3) if $c\mid ab$, then $c\mid a$ or $c\mid b$. Send a solution
Problem 4. Formulate divisibility rules for the following numbers 1) 2, 2) 3, 3) 4, 4) 5, 5) 9, 6) 11. Send a solution
Problem 5. Consider a number such that the sum of its digits is 2004. Is it possible that this number is a square of an integer number? Send a solution
Problem 6. * A number is three times bigger than the sum of its digits. Show that this number is divisible by $27$. Send a solution
Problem 7. Show that 1) if $a^2 ⋮ (a+b)$, then $b^2 ⋮ (a+b)$; 2*) if $x+y+z\ne0$, then $(x^3+y^3+z^3-3xyz) ⋮ (x+y+z)$. Send a solution
Problem 8. Which numbers have an odd number of positive divisors? Send a solution
∆ Definition 2. A number $p>1$ is called prime number if it is divisible only by $1, -1, p, -p$. Otherwise, a number $p>1$ is called a composite number.
Problem 9. Show that there are infinitely many prime numbers. Send a solution
Problem 10. Show that for any $n$ we can find $n$ consecutive composite numbers. Send a solution
Problem 11. * Denote by $n?$ the product of all prime numbers smaller than $n$. Show that for $n>3$ we have $n?>n$. Send a solution
Problem 12. 1) Find all prime numbers $p$ such that $p+2$ and $p+4$ are also prime. 2*) Show that there are infinitely many prime numbers $p$ such that $p+2$ is also prime. Send a solution
Problem 13. (the Sieve of Eratosthenes) Numbers from $2$ to $1000$ are written on a blackboard. Eratosthenes circles number $2$ and erases all numbers other than $2$ which are divisible by two. Then he repeats the process: he circles the smallest not yet circled number and erases all others which are divisible by it (doesn't matter if they are circled already). He stops once only circled numbers are left on the blackboard. What numbers are left on the blackboard? Send a solution
Problem 14. Write down all prime numbers that are smaller than $100$. Send a solution
Problem 15. Show that number $a$ is composite if and only if $a$ is divisible by a prime number that is at most $\sqrt{a}$. Send a solution
Problem 16. Show that 1) any number that is bigger that $1$ can be expressed as a product of prime numbers; 2) a number $x>1$ can be expressed as \begin{equation*} x=p_1^{a_1}p_2^{a_2}\ldots p_n^{a_n}, \end{equation*} where $p_1< p_2< \ldots< p_n$ are prime numbers and $a_1, a_2,\ldots, a_n$ positive integer numbers. Such expression of a positive integer is called the canonical representation; 3*) (Fundamental theorem of arithmetic) if $x$ is expressed in the following ways \begin{equation*} x=p_1^{a_1}p_2^{a_2}\ldots p_n^{a_n}= q_1^{b_1}q_2^{b_2}\ldots q_m^{b_m}, \end{equation*} then $m=n$, and for any $1\leqslant i\leqslant n$ $p_i=q_i$, $a_i=b_i$; 4) if in the above expression each $a_i$ is even, then $x$ is a square of some number, i.e. there is $y$ such that $x=y^2$. Send a solution
Problem 17. Find the canonical representations of the following integers $1024$, $57$, $84$, $91$, $391$, $101$, $1000$, $1001$, $1543$. Send a solution