# Tagged Questions

Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $b-a$.

31 views

### Show that, of any set of $2^{n+1}-1$ positive integers numbers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$.

Show that, of any set of $2^{n+1}-1$ positive integers numbers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$. My approach: Let $a_1,\ldots,a_{2^{n+1}-1}$ positive ...
24 views

10k views

### Calculator model with mod function?

I'm wondering does anyone know of a scientific calculator with a mod function? In C# this is shown as follows (just in case there are any other mods that a mathematical non-savant such as myself ...
43 views

53 views

### Why can we exchange numbers when working with modulo expressions?

Please excuse me if the answer is obvious because I'm a beginner. Why can we exchange numbers when working with modulo expressions? For example: $$4^2 \equiv (-1)^2 \pmod{5}$$ You may say the ...
45 views

### Show that the congruence $3x^2 \equiv 12 \pmod{12}$ has a solution, or not [on hold]

Someone know Quadratic residues ? Below: Show that the congruence $3x^2 \equiv 12 \pmod{12}$ has a solution, or not.
327 views

### Prove: $a\equiv b\pmod{n} \implies \gcd(a,n)=\gcd(b,n)$ [duplicate]

Proof: If $a\equiv b\pmod{n}$, then $n$ divides $a-b$. So $a-b=ni$ for some integer $i$. Then, $b=ni-a$. Since $\gcd(a,n)$ divides both $a$ and $n$, it also divides $b$. Similarly, $a=ni+b$, and since ...
35 views

### Does there exist a general technique for solving systems of multivariable linear congruences

I'm aware for coprime moduli we have the CRT for solving the problem $$\begin{matrix} a_0 x \equiv b_0 \mod m_0 \\ a_1 x \equiv b_1 \mod m_1 \\ \vdots \\ a_n x \equiv b_n \mod m_n \end{matrix}$$ ...
### What will be the multiplicative inverse of square root of 5 with respect to a natural number $M$?
Can such a number $N$ be found such that $\sqrt{5}N \equiv 1 \mod M$? If no,what can be the best approximation for $N$?
Let $p$ be an odd prime. Given that $a\equiv b \pmod p$ and $c \equiv d \pmod p$, such that none of $a,b,c,d$ is a multiple of $p$. Under what conditions, $\frac{a}{c} \equiv \frac{b}{d} \pmod p$.