# Tagged Questions

7 views

### If there is a subset with sum divisible by n, then take out an integer of the subset. How many moves?

Fix an integer $n \ge 2$. A finite set $A \subset \mathbb{N}$ is given. Define $s(X) = \sum_X x$, where $X$ is a finite set. We know that $n \mid s(A)$. We can do just one move: if there is a ...
48 views

### Combinatorial interpretation of an equality

In a recent project, I came up with the following equality which turned out to be extremely useful for counting conjugacy classes in certain division algebras (I won't go into the details here, it's ...
45 views

### To prove that ${2+i \choose i}\equiv k \mod n$ is not possible that $k=0,1,\ldots,n-1 \forall i\ge 0$ and $i \in \mathbb{Z}$ and $n$ is odd.

To prove that ${2+i \choose i}\equiv k \mod n$ is not possible that $k=0,1,\ldots,n-1 \forall i\ge 0$ and $i \in \mathbb{Z}$ and $n$ is odd. This is a problem from ISI 2014 written test in a little ...
33 views

### Given any integers $a,b,c$ and any prime $p$ not a divisor of $ab$, prove that $ax^2+by^2\equiv c\pmod{p}$ is always solvable.

The fact that there are $\dfrac{p+1}{2}$ quadratic residues seem to me to help solving the question, but I don't know how to go on from that point. Could you give me any hint?
72 views

### Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$. This is a problem from a selection to IMO 2014. I like thinking about this problem, it is ...
30 views

How many solutions does the following equation have: $x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}\equiv0\mod11$ where $x_{1...9} \in \{0,1,2,3,4\ ...\ 8,9\}$ and $x_{10}\in\{0,1,2,3,4\ ... 1answer 201 views ### Fermat's little theorem for$n=3$for$N > 0$, I'm trying to show Fermat's little theorem, for$3$using the orbit stabilizer theorem:$N^3 - N$an element of$3\mathbb{Z}\ (3 \mod \mathbb{Z})$Pf/ we can break it down into ... 3answers 67 views ### Prove how many distinct elements in the set$\{ax \pmod{m}:a\in\{0,…,m-1\}\}$There are$\dfrac{m}{\gcd(m,x)}$distinct elements in the set$\{ax \pmod{m}:a\in\{0,...,m-1\}\}\$ I have only known these by using a computer to generate the number of distinct elements. But I am not ...
I am working on a programming problem where I need to calculate 'n choose k'. I am using the relation formula $${n\choose k} = {n\choose k-1} \frac{n-k+1}{k}$$ so I don't have to calculate huge ...