Problems

The number $a$ has a prime factorization $2^3\times 3^2\times 7^2\times 11$. Is it divisible by $54$? Is it divisible by $154$?

a) The number $a$ is even. Should $3a$ definitely also be even?

b) The number $5c$ is divisible by $3$. Is it true that $c$ is definitely divisible by $3$?

c) The product $a\times b$ is divisible by $7$. Is it true that one of these numbers is divisible by $7$?

d) The product $c\times d$ is divisible by $26$. Is it true that one of these numbers is divisible by $26$?

a) The number $a^2$ is divisible by $11$. Is $a^2$ necessarily also divisible by $121$?

b) The number $b^2$ is divisible by $12$. Is $b^2$ necessarily also divisible by $144$?

What is the smallest integer $n$ such that $n\times (n-1)\times (n-2)...\times 2$ is divisible by $990$?

Jack believes that he can place $99$ integers in a circle such that for each pair of neighbours the ratio between the larger and smaller number is a prime. Can he be right?

a) Prove that a number is divisible by $8$ if and only if the number formed by its laast three digits is divisible by $8$.

b) Can you find an analogous rule for $16$? What about $32$?

Look at this formula found by Euler: $n^2+n+41$. It has a remarkable property: for every integer number from $1$ to $21$ it always produces prime numbers. For example, for $n=3$ it is $53$, a prime. For $n=20$ it is $461$, also a prime, and for $n=21$ it is $503$, prime as well. Could it be that this formula produces a prime number for any natural $n$?

Denote by $n!$ (called $n$-factorial) the following product $n!=1\cdot 2\cdot 3\cdot 4\cdot ...\cdot n$. Show that if $n!+1$ is divisible by $n+1$, then $n+1$ must be prime. (It is also true that if $n+1$ is prime, then $n!+1$ is divisible by $n+1$, but you don’t need to show that!)

a) In a canteen, every day a chef prepares three lunch options customers can choose from. He is not a very good chef, but he knows six meals he can prepare very well. Every day, he chooses three out of these six and offers them. The options are presented left to right and we consider a lunch different if the three options are in different order, even if they are the same. For how many days can the chef go on, without repeating himself?

b) The customers have seen through chef’s plot and they realized that the order of the options does not in fact matter – there are still the same three lunches to choose from. If the chef now wants every day to be different, for how many days can he prepare different three meals each day?

A magician has $10$ ingredients used for brewing potions. Any $6$ have to be combined in order for brewing to be successful. How many different potions can the magician brew?