Linked Questions

-1
votes
1answer
356 views

Multiplicative Inverse for Elements in an Integer Ring [duplicate]

In the book I'm reading it says that there exist elements with multiplicative inverses in a integer ring if $gcd(a,m)=1$ where $a$ is the element and $m$ is the modulo. The inverse $a^{-1}$ is ...
0
votes
1answer
168 views

Solving congruence system with no multiplicative inverse [duplicate]

I am trying to find a way of solving congruence systems of the form: $$ b*x = a \quad mod \quad y $$ Where $b$ and $y$ are not prime to each other. My current way of solving congruence systems ...
-1
votes
1answer
73 views

Number of solutions to $89 x \equiv 7 \pmod{55}$ [duplicate]

How do I show that my solution to the congruence is the only solution? I'm asked to solve, $\mu,\lambda$ that satisfy $$89 \lambda+55 \mu=1$$ Using Euclid's algorithm I found $\lambda=-21,\mu=34$. ...
0
votes
2answers
42 views

Modulo arithmetic: finding “c” when result of right-hand side expression is a float [duplicate]

In my series of questions on modular arithmetic, I stumbled upon cases where normal textbooks don't explain much. Now my problem is to find $c$ when the expression to be solved (right-hand side) ...
-1
votes
1answer
56 views

How many solutions are there to the equation $na\equiv _m0$ for $0\leq a <m$? [duplicate]

I'm reading trough a proof that the number of (group) homomorphisms $\mathbb{Z}_n\rightarrow \mathbb{Z}_m$ is $$\text{gcd}(n, m),$$ and this is the only step that I'm not understanding, namely, that ...
0
votes
1answer
29 views

If $x\equiv a\pmod{n}$, prove that either $x\equiv a\pmod{2n}$ or $x\equiv a+n\pmod{2n}.$ [duplicate]

If $x\equiv a\pmod{n}$, prove that either $x\equiv a\pmod{2n}$ or $x\equiv a+n\pmod{2n}.$ Well I am a bit stuck in this congruence problem. I tried out :- $n\vert x-a$ so for $2n|x-a$, $x-a$ must be ...
1
vote
0answers
33 views

Find the solutions to $4x \equiv 16 \pmod{20}$ [duplicate]

Find the solutions to $4x \equiv 16 \pmod{20}$ I managed to determine this by writing $4x = 16 + 20j \implies x =4 +5j \implies x \equiv 4 \pmod{5}$, but the official solution stated that the ...
0
votes
1answer
45 views

How to do division of two numbers which are already under modulo 'm'? [duplicate]

How to do division for the following example? Case 1 : Without modulo n1 = 40, n2 = 8 Quotient = n1/n2 = 5 Case 2 : With modulo m = 6 n1 = n1 mod m = 4 (AND) n2 = n2 mod m = 2 Quotient = 4 / 2 =...
8
votes
7answers
1k views

Solving $ax \equiv c \pmod b$ efficiently when $a,b$ are not coprime

I know how to compute modular multiplicative inverses for co-prime variables $a$ and $b$, but is there an efficient method for computing variable $x$ where $x < b$ and $a$ and $b$ are not co-prime, ...
10
votes
3answers
1k views

mod Distributive Law, factoring $\!\!\bmod\!\!:$ $\ ab\bmod ac = a(b\bmod c)$

I stumbled across this problem Find $\,10^{\large 5^{102}}$ modulo $35$, i.e. the remainder left after it is divided by $35$ Beginning, we try to find a simplification for $10$ to get: $$10 \equiv 3 ...
8
votes
2answers
10k views

Modular Fraction Arithmetic

I just want to confirm I am doing this problem correctly. The problem asks to compute without a calculator: $$ 3 * \frac{2}{5} \pmod 7 $$ The way I am solving the problem: $$ 3 * \frac{2}{5} \bmod 7 \...
6
votes
3answers
1k views

Existence & uniqueness of solutions of linear congruences $\ cx\equiv b\pmod m$

Prove that one can solve the congruence $cx \equiv b \pmod m \Longleftrightarrow \gcd(c,m)|b$. Show, moreover, that the answer is unique $\bmod{m/\gcd(c,m)}$ My Work Proof of $(\Rightarrow)$: ...
-1
votes
4answers
94 views

$12x \equiv 33 \pmod{57}$ [closed]

Can you show me a step by step instruction how to solve this problem? $$12x \equiv 33 \pmod{57}$$
0
votes
5answers
60 views

Solve the linear congruences

To solve $\,26x\equiv 6\pmod{\!110},\,$ note $\,\gcd(26,110)=2\,$ so there are $\,2\,$ solutions. Next, $\ 104x\equiv 24\pmod{\!110}\ $ follows by scaling the above by $\,4.$ i.e. $\,\ \ \ \ \ {-}6x\...
1
vote
1answer
49 views

Name of Modulus Theorems

Let p and q be two positive integers with r being their greatest common factor. Consider the set E such that $E = \{0,\:\:r, \:\:2r,\:\:3r,\:\: ...,\:\: (q-r)\}$ Let an be a sequence defined for ...

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