0
votes
2answers
40 views

Why 3 is a multiplicative inverse of 7 in modular arithmetic?

why is 3 a multiplicative inverse of 7 in modular arithmetic of 5 ? I'm not able to understand how this is true. PS: I know 3*7-1 % 5 = 0. I'm not able to make sense of inverses in modular ...
7
votes
1answer
74 views

$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $

Show that $$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $$ Indeed, First let's show $7\mid x \text{ and } 7\mid y \Longrightarrow 7\mid x^2+y^2 $ we've $7\mid x ...
0
votes
1answer
28 views

Can we define the equality as $a=b$ iff $\frac{a}{b}=1$?

Well, The title i guess is enough to get what i'm looking for: I'm wondering if we can define equality of let's say $a$ and $b$ that the devision of $a$ over $b$ or $b$ over $a$ is $1$ : $$a=b ...
0
votes
1answer
50 views

Remainder of $1946^{1972} : 26$

Is this correct? $1946^1 = 22 \mod{26}$ $1946^2 = 22^2 = 484 = 16\mod{26}$ $1946^3 = 22^2 * 22 = 16 * 22 = 14 \mod{26}$ $1946^4 = 22^2 * 22^2 = 16^2 = 22 \mod{26}$ And therefore for any integer ...
0
votes
1answer
23 views

Explanation and validation of point adding/doubling on elliptic curves

I'd like to implement point multiplication on elliptic curves over prime fields. My problem is that I've found different definition how to do it. At adding: the second parameter of the result is not ...
1
vote
0answers
61 views

Why doesn't this base 10 number x mod 2^y work for converting base 10 to binary

Okay I tried to convert 1 million to binary by dividing by a power of 2 and taking the remainder and dividing that by a power of 2 and so on and I got this: 1111010000100100000 Google says 1 million ...
0
votes
2answers
28 views

Modular arithmetics

I am having a question specific for this video on Youtube: https://www.youtube.com/watch?v=3QnD2c4Xovk They seem to explain that shared encryption concept pretty well, but I seem to get some incorrect ...
1
vote
1answer
53 views

division and remainders

When you divide, it will not always result in whole number. Sometimes there will be numbers left over. You can either end the problem with a remainder or use decimal points to get a decimal number. ...
0
votes
0answers
47 views

number of cubes $mod\space p$

I was wondering if there is an analogue, for cubes, of the fact that half of the elements of $(\mathbb Z/p\mathbb Z)^*$ are squares. I checked with a program, and it seems that whenever $p-1$ is ...
1
vote
1answer
41 views

Determine $2$ missing digits for modulo $11$

An account number verification system works as follows: All digits in a 10-digit account number all multiplied by following weights: $$(6 \ 3 \ 7 \ 9 \ 10 \ 5 \ 8 \ 4 \ 2 \ 1)$$ Resulting numbers ...
2
votes
3answers
64 views

Finding the remainder when $1.1!+2.2!+3.3!+ … +10.10! +2$ is divided by $11!$

Find the remainder when $1.1!+2.2!+3.3!+ ... +10.10! +2$ is divided by $11!$ An attempt: Rearranging: $$\frac{1}{11!}+\frac{2.2!}{11}+\frac{3.3!}{11} \cdots +\frac{10.10!}{11}+\frac{2}{11!}$$ ...
0
votes
1answer
34 views

Fractional exponentiation in modular arithmetic

Does raising a modular expression to a fraction mean anything? For example, $a\,\,mod \,\,N$ raised to $1/b$ where $b>0$. Does this violate the rules of modularity?
5
votes
1answer
56 views

$x;y;z\in \mathbb{Z}$ such that $(x-y)(y-z)(z-x)=x+y+z$. Prove that : $27\mid x+y+z$

Let $x;y;z\in \mathbb{Z}$ such that $(x-y)(y-z)(z-x)=x+y+z$. Prove that : $27\mid x+y+z$ Thanks :) P/s : I don't have any ideas about this problem..!!
3
votes
2answers
95 views

Is there a name for sequences like these?

Starting from an integer value (say $0$ in these cases), I need a sequence of integers to add in a cycle that progress through the integers visiting each exactly once. For example, the most obvious ...
0
votes
1answer
118 views

Is $2^k = 2013…$ for some $k$? [duplicate]

I'm wondering if some power of $2$ can be written in base $10$ as $2013$ followed by other digits. Formally, does there exist $k,q,r \in \mathbb N$ such that $$2^k=2013 \cdot 10^q+r \,\,\,; ...
1
vote
2answers
289 views

Remainder when $26^{3008} + 3008^{26}$ is divided by $4$

I want to find the Remainder when $26^{3008} + 3008^{26}$ is divided by $4$. What should I do? Even though I've included the tag modular arithmetic I've very limited knowledge about it. How should I ...
0
votes
1answer
42 views

A proof of existence

I need help proving the following result. Let $p$ be a positive prime number $\geq 5$, $x$ a non zero integer and $y$ a non zero positive integer such that $x^2-y^p=1$ I've successfully proved that ...
4
votes
3answers
328 views

Concepts of Modern Mathematics (Ian Stewart) - 751=7.107+2?

Concepts of Modern Mathematics by Ian Stewart (1995). In Chapter 3 Ian Stewart talks about Short Cuts in the Higher Arithmetics, one section is on modular arithmetics. When talking about the days of ...
3
votes
2answers
91 views

Two questions on clock arithmetic

I have two questions on clock arithmetic, both of which I have solved, but I am looking for neater proofs. Let us suppose we have a circle named $\mathbb{Z}_n$ with $n$ equally spaced points on it ...
3
votes
4answers
200 views

Prove that $2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$

I am utterly new to modular arithmetic and I am having trouble with this proof. $$2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$$ It's because $2+5=3+4=7$, but it's not so clear for me ...
1
vote
1answer
469 views

Reverse mod operation getting bounded number

Is it possible to get the reverse of the mod operation if I just want the first possible number? I mean, if I can bound the initial number. For example: I want to do $(x+y) \pmod {10} = z$ ($x$ ...
1
vote
0answers
80 views

Need help simplifying an equation.

I'm trying to speed up the following code: sum = 0 for (k = 1 ... N) { f = Fibonacci(k); for (a = 1 ... 24) for (b = 1 ... 24) for (c = 1 ... 24) { sum = sum + m(a, b, c) // ...
0
votes
1answer
953 views

how many 2x2 matrices are invertible in mod p

I am trying to solve this problem for homework but unable to get anything. The question is to find the number of invertible 2x2 matrices in mod p? Each entery can bee from the set ...
5
votes
3answers
523 views

How to calculate $3^{45357} \mod 5$?

I wrote some code, here is what it gives: \begin{align*} 3^0 \mod 5 = 1 \\ 3^1 \mod 5 = 3 \\ 3^2 \mod 5 = 4 \\ 3^3 \mod 5 = 2 \\\\ 3^4 \mod 5 = 1 \\ 3^5 \mod 5 = 3 \\ 3^6 \mod 5 = 4 \\ 3^7 \mod 5 = 2 ...
2
votes
4answers
169 views

How to implement modular division?

I want to calculate do calculate $\frac{a}{b} \pmod{P}$ where $a$ is an integer less $P$ and $P$ is a large prime, $b=5$, a fixed integer. How can I implement it? Any algorithm which will work well?
3
votes
2answers
808 views

Can somebody simply explain Wilson's theorem (for a 13 year old)

I am Rohan Kapur. This is my first time posting on the Mathematics site, although I am quite active on StackOverflow, the programming site. I am doing a Islamic Maths assignment at the moment for ...
1
vote
0answers
101 views

Collaborative modular exponentiation

EDIT: Rephrased. I have, stored somewhere, the values $a$ , $Q$, $N_1$ (plus its factor) and $a^{2Q} \mod N_1$. I also know $b$, $R$ and $N_2$ (but not its factors). I want to know whether there is ...
1
vote
1answer
150 views

A rule to determine the crossed out digit

Lets take any integer, $z=abc\cdots$, form the sum of its digits, $a+b+c+\cdots$, subtract this from $z$, cross out any one digit from the result, and denote the sum of the remaining digits by $w$. ...
4
votes
5answers
282 views

What is the remainder of $(14^{2010}+1) \div 6$?

What is the remainder of $(14^{2010}+1) \div 6$? Someone showed me a way to do this by finding a pattern, i.e.: $14^1\div6$ has remainder 2 $14^2\div6$ has remainder 4 $14^3\div6$ has remainder 2 ...
2
votes
2answers
773 views

Schönhage-Strassen multiplication

I am trying to implement the Schönhage-Strassen algorithm (SSA) for multiplying large integers, but it only gives the right result if all $\delta_j$ are zero. I'll explain what I mean by this: SSA ...
1
vote
1answer
105 views

Casting nines in other bases (radix) and doing check sums for binary

I was showing my son how to cast out nines the other day. He noted that based on the way it worked, we should be able to cast out 7s when we work with octal. We tested this in several bases and it ...
2
votes
3answers
1k views

How to do addition/subtraction with n complement eg. Decimal subtraction via 10s complement

How do I do in general addition/subtraction with n's complement? With binary numbers its straightforward, convert the number you want to subtract to 1's or 2's complement then add the numbers. Handle ...
12
votes
2answers
692 views

nth powers modulo all primes

Let $a \in \mathbb{Z}$, $n \in \mathbb{N}^*$ be integers, and set $P=X^n - a$. Let us consider the three following statements : 1) $P$ has a root in $\mathbb{Z}$ (i.e. $a$ is an nth power) 2) $P$ ...