Questions tagged [modular-arithmetic]
By Gabriel Cooper •
Ask Question
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 $a-b$.
12,212 questions- Bountied 0
- Unanswered
- Frequent
- Score
- Unanswered (my tags)
Prove that the linear congruence $ax+by \equiv c \mod m$ has exactly $m \times \gcd(m,a,b)$ many solutions $(x,y)$ satisfying $0 \le x,y \le m-1$. [closed]
Let $a,b,c$ and $m$ be positive integers. $(a)$ Prove that the linear congruence $ax \equiv c \mod m$ has either no solution or it has exactly $ \gcd(m,a)$ many solutions $x$ satisfying $0 \le x \le m-... elementary-number-theory modular-arithmetic- 151
Deduce that $\{a_0,a_1,\cdots,a_{m-1}\}$ is not a complete set of residues $\mod m$.
Let $m$ be a positive integer. Define $a_i = \frac{i^2+i}{2}$ for $i \ge 0.$ $(a)$ Let $m = 2^n$ for a nonnegative integer $n$. Prove that $\{a_0,a_1,\cdots,a_{m-1}\}$ is a complete set of residues $\... elementary-number-theory modular-arithmetic- 151
Can someone explain these groups of linear patterns in the dropping times of Collatz Sequences? Could this lead to a proof?
Please buckle in because this may be a long post, but I think it will be necessary to help the reader understand three things: How this data was generated. How the data is grouped into different '... number-theory modular-arithmetic dynamical-systems collatz-conjecture music-theory- 123
Solving a big quadratic congruence.
Is there a way of solving $x^2 \equiv 1156 \text{ }(\text{ mod } 3^2 5^3 7^5 11^6)$ without having to solve 16 linear simultaneous congruences? I have found the solutions $x\equiv 34 \text{ } (\text{ ... elementary-number-theory modular-arithmetic- 60
How to make addition modular of real number
I think a question, this question just rised into my head.(I don't know but I'm feel this stupid question) In $\mathbb{Z}_n$ group, we have a properties of modular arithmetic. a , b $\in \mathbb{Z}_n$ ... group-theory discrete-mathematics modular-arithmetic ceiling-and-floor-functions- 1
There exists a subsequence of every $n$ consecutive natural numbers whose sum is divisible by $n(n+1)/2$
This is my first question here, hence I apologize beforehand for any mistakes I make. The statement is very simple. For any natural $n$, show that for all sequences of $n$ consecutive natural numbers,... combinatorics modular-arithmetic- 65
Solving system of congruence [duplicate]
i have to solve : $33x ≡ 24 [45]$ and $30x ≡ 6[72]$ by Euclid Algorithm : $x ≡ -2 [15]$ and $x ≡ 5 [12]$ $pgcd(15,12) = 3$, i don't know how to conclude modular-arithmetic arithmetic- 1
How can I solve $\sum\limits_{n=1}^m (5 k_{n}+7) \mod 8$? [closed]
$$\sum\limits_{n=1}^m (5 k_{n}+7) \mod 8$$ How can I solve this? modular-arithmetic- 1
What does "$A^b \bmod c$, where $A$ is a square matrix" mean? What is the modulus of a matrix?
I was reading Wikipedia's "Modular exponentiation" entry. It made sense to me until I got to the part about Matrices. What does "$A^b \bmod c$, where $A$ is a square matrix" mean? ... matrices modular-arithmetic- 429
How would you approach this modulus question? [closed]
How would you approach this modulus question: $$\begin{cases} 35x+36y \equiv36 \bmod {39} \\ 20x+18y \equiv2 \bmod {33} \end{cases}$$ I know that I have to make the $\text{mod}$ the same, but I don't ... elementary-number-theory modular-arithmetic- 1
Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and for every $[a]_{n}$ we get $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ .
Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and every $[a] \in \mathbb{Z}_{(m,n)}$ can be regarded as an element $[a]_{n}$ such that $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ . ... abstract-algebra group-theory ring-theory modular-arithmetic modules- 85
Egyptian unit decomposition + coverings of $\mathbb{Z}_P$ by arithmetic progressions
For a quick exposition of how I arrived at this question: I found the Wikipedia page on covering sets and through some experiments on my own (fixing the constant at the end, the factor $k$ itself etc.)... elementary-number-theory reference-request modular-arithmetic arithmetic-progressions egyptian-fractions- 396
can it be proven that $4^n \mod 15$ is equal to 1 for even n, 4 for odd n? [duplicate]
Can it be proven that $4^n \mod 15$ is equal to 1 if $n$ is even, 4 if $n$ is odd? it seems like this holds for fairly high $n$'s, but I was unable to rigorously prove this statement. I tried to state ... modular-arithmetic- 55
How to find $40! \mod 5^{10}$?
I'm trying to solve this congruence. I found that in $40!$ there are $9$ powers of $5$, but I'll need one more for $40!=0 \mod 5^{10}$. I thought about Euler's theorem or Wilson's, but couldn't find ... elementary-number-theory modular-arithmetic- 11
Can we generalize the quadratic formula to modular arithmetic?
Does the quadratic formula $\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ hold modulo $n$ for $ax^2 + bx + c \equiv 0 \pmod n$? Computing the square root would require factoring $n$ and using ... modular-arithmetic quadratics quadratic-residues- 237
15 30 50 per page12345…815 Next
