# 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 $b-a$.

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### A add/subtract function that rotates numbers from 1 to 12

I'm having difficulty searching for this since I don't know what it's called. I want a function which will add/subtract in a circular fashion from 1 to 12. I could do this with logic operators (if ...
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### Proof of: If $a \equiv b \pmod{d}$ and $x \equiv y \pmod{d}$ then $a + x \equiv b + y \pmod{d}$ and $ax \equiv by \pmod{d}$

I am trying to learn mathematics from the beginning, i.e. trying to form a solid foundation and understanding of basic concepts that I should have learned in high school. I am working through Basic ...
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### Prove that 8 is the remainder of $5^{336}$ by $23$

I've searched this website and while there are a few questions similar to mine, I couldn't find what I was looking for/a specific method for what I want to do. I want to understand how one would ...
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### Primitive roots and 'equivalent exponents'.

If M is a primitive root mod p and M = $\ N^T$ mod p , then the order of N mod p is also (p-1) is this true?
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### Show that there exists no integer $x$ such that $3x$ is congruent to 5 (modulo 6)

So far my approach was to rewrite the congruency to $5-3x=6t$ for some integer $t$. My problem is I get stuck in trying to show how $5-3x$ is never divisible by $6$.
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### Equation system modulo prime

I have an excercise, it is to solve $$9\equiv_{p}8k_1+k_2$$ $$32\equiv_{p}6k_1+k_2$$ $$45\equiv_{p}11k_1+k_2.$$ $k_2$ is easily eliminated from the equations but I don't know how to proceed from ...
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### How does the Euler Totient Function apply here?

How many positive integers $< 2013$ are divisible by $2$ Can I somehow use Euler's Totient function to find this?
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### Euler's theorem (modular arithmetic) for non-coprime integers

I am trying to calculate $10^{130} \bmod 48$ but I need to use Euler's theorem in the process. I noticed that 48 and 10 are not coprime so I couldn't directly apply Euler's theorem. I tried breaking ...
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### Prove that if a solution exists to the congruences $x \equiv a$ (mod $n_1$), $x \equiv b$ (mod $n_2$), then it is unique modulo lcm($n_1, n_2$)

Prove that if a solution exists to the congruences $x \equiv a$ (mod $n_1$), $x \equiv b$ (mod $n_2$), then it is unique modulo lcm($n_1, n_2$) I'm having a trouble showing this. I think I need to ...
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### Solve diophantine using modulus

Find all pairs of positive integers $(m, n)$ that satisfy, $mn + 3m - 8n = 59$ Using Modular arithmetic. Okay, this is a diophantine equation, where can I begin?