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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Units digit when there is a power of power

How do you find the units digit in case of an expression like this $$ 7^{8^7} $$ I know how to find the units digit when there is one integer and there is only one power. But how do I find it when ...
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4answers
42 views

How to manually determine big number congruences

How is it possible to determine if the the following congruence is true manually, with resort to a basic calculator? The real problem here is how to do the math with a such big number? $$ 2015^{50} ...
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1answer
88 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
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1answer
38 views

Polynomial function residues

If we use Euclid's representation for integers $n=aq+r$, we can write $n\equiv r \mod q$. We can also write functions similarly, for example $n(x)=a(x)q(x)+r(x)$ and so I imagine we can write ...
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0answers
19 views

One dimensional representations of $SL_2(\mathbb{Z}/n\mathbb{Z})$

Someone knows a reference or knows how to calculate the linear character of $SL_2(\mathbb{Z}/n\mathbb{Z})$, for an arbitrary $n$? Thanks
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4answers
34 views

system of modular equations.

$x\equiv 2\pmod3$ $x\equiv 3\pmod 5$ $x\equiv 7 \pmod{11}$ How can I solve this system for $x$? I've tried all kinds of things using divisibility but no success. Any hints of solutions are greatly ...
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3answers
42 views

Smallest divisible repunits

A repunit of length k is a number containing k ones (1, 11, 111...). R(k) is defined to be the repunit of length k. A(n) is the least value of k such that R(k) is divisble by n (assuming gcd(n, 10) ...
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6answers
226 views

Calculating remainder of $666^{666}$ when divided by $1000$.

I want to calculate the remainder of $666^{666}$ when divided by $1000$. But for the usual methods I use the divisor is very big. Furthermore $1000$ is not a prime, $666$ is a zero divisor in ...
4
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1answer
66 views

For any $n$ positive integers ($n\geq 5$) exactly 3 or 4 of them are equal to each other modulo $2^m$ for some $m$

How can one prove that for any $n$ distinct positive integers, $n\geq 5$, there exists $m$ such that exactly 3 or 4 of them are equal to each other modulo $2^m$? I tried to prove it for small $n$. ...
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27 views

Does this modulo formula change odds?

Good day, Consider 5 people each choosing a different number between [0 and A] randomly At the end, we add all the numbers: $n_1+n_2+n_3+n_4+n_5 =N$ Then modulo A+1: $W = N \pmod{A+1}$ My ...
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1answer
20 views

Intervals of integers modulo n

Do the following related concepts appear anywhere in literature? Denoting an "interval" in the integers modulo $n$ by $[i,j] = \{i, i+1, \dotsc, j\}$. For example, in modulo 6, $[5,3] = ...
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5answers
73 views

How can I prove that $4^{2012} \mod 8$ is $0$

Prove that $4^{2012} \mod 8 = 0$ I'm not really sure what rule I should use to prove this.
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4answers
82 views

Solving $x^2=17\pmod{128}$

I'm attempring to solve a congruence $x^2 \equiv 17\pmod{128}$ but not quite sure how to go about it. I see that $128 = 2^7$, but the Chinese Remainder Theorem doesn't apply to $\gcd > 1$. I found ...
2
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1answer
51 views

Modulo calculation with multiple exponents, via CRT

I'm aware that there are already a few questions like this but unfortunately I wasn't able to find an answer yet. $$ (14^{2014)^{2014}} \pmod {60} $$ So I started off by putting the modular in : $$ ...
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3answers
34 views

Double modular exponent with Euler-Fermat

$$7^{3^{18}} \pmod{9}$$ Using this formula : $a^{\phi(m)} \equiv 1 \pmod m$ I got $7^6 \equiv 1 \pmod{9}$ and I can write $3^{18}$ as $3^6 \cdot 3^3$ And what are next steps? I got stuck here.
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2answers
57 views

$9^{123456789} \pmod{100}$ , retrace calculation operation

I have $9^{123456789} \pmod {100}$. I did use Euler's Theorem, and got $\phi = 40$ and therefore I can say $$9^{123456789 \pmod {40}} \pmod {100}$$ this equals $ 9^{29} \pmod {100}$. Then in one of ...
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1answer
46 views

If p is an odd prime, show that $p^2 \equiv 1 \pmod 8 $

If p is an odd prime, show that $p^2 \equiv 1 \pmod 8 $. I know that odd numbers are of the form $2k \pm 1$. Then $p^2=(2k \pm 1)^2= 4k^2 \pm 4k +1$. But it does not help to solve.
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2answers
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Is this problem correct? [duplicate]

I have found another problem in my book. I have to prove that $$2^{70}+3^{70}$$ is divisible by 13. But I have proven that $2^{70}\equiv 12 (mod 13)$ and $3^{70}\equiv 3 (mod 13)$ so it is ...
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2answers
64 views

How can I find the remainder?

How can I find the remainder when $$(12371^{56}+34)^{28}$$ is divided by $111$. I have tried congruences modulo $111$ but without any success.
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1answer
52 views

Why is $x^{100} = 1 \mod 1000$ if $x < 1000$ and $\gcd (x,1000) = 1$?

Let $U(1000) =$ the multiplicative group of all integers less than and relative prime to $1000$. "Show that for every $x \in U(1000)$ it is true that $x^{100} = 1 \mod 1000$." Been thinking ...
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3answers
46 views

What is the relation between the linear combination and modular arithmetic?

What is the relation between the linear combination and modular arithmetic? The linear combination is in a field and there must be some fundamental relation between them. What is it?
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3answers
43 views

Is it true that $x \nmid (q-1) \implies 2^x \not \equiv 1 \mod q$

If $q$ is a prime number, then from little fermat theorem it is known that $$2^{q-1} \equiv 1 \mod q$$ My doubt is that If $x \nmid (q-1)$ then $2^x \not \equiv 1 \mod q$ is true statement or not? ...
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30 views

binary representation of integers congruent 1 and 3 modulo 4

Let $k=b_nb_{n-1}\ldots b_3b_2b_1b_0$ be the binary representation of an odd positive integer. Prove: If $k\equiv 1 \mod 4$ then $b_1=0$. If $k\equiv 3 \mod 4$ then $b_1=1$. I think that to prove ...
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3answers
55 views

Let $z \in \Bbb Z_m$, when is $z^2 \equiv 1$?

Let $z \in \Bbb Z_m$. When is $z^2=1, (z\neq1)$? I know that for $m$ prime, $z=p-1$ is it's own inverse, but what about nonprime $m$? Is $p-1$ the only self inverse element in $\Bbb Z_p$ ?
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2answers
56 views

Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$? [closed]

Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$ in modular arithmetic?
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3answers
57 views

Remainder when divided by 9

I'd like help with this question : What is the remainder when $$2^{2} + 22^{2} + 222^{2}+ \ldots + \underbrace{2222...22^{2}}_{49 \text{ times}} $$ is divided by $9$
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6answers
127 views

What are the last two digits of $77^{17}$?

I'm trying to solve current task referenced the following but I stuck at $\displaystyle77^{17}\equiv x\pmod{100}$. As it is described on above link it uses Binomial Theorem. But I read a lot about the ...
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Generalizing the Pell equation $x^2-61y^2 = 1$

In a table of fundamental solutions $f_1(x,y)$ to Pell equations, $$x^2-dy^2=1\tag1$$ with $d<110$, two will stand out, $$(U_{61})^6 = \big(\tfrac{39+5\sqrt{61}}{2}\big)^6 = x+y\sqrt{61} ...
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2answers
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$x^2 + 3x + 7 \equiv 0 \pmod {37}$

I'm trying to solve the following $x^2 + 3x + 7 \equiv 0 \pmod {37}$ What I've tried - I've tried making the left side as a square and then I know how to solve but couldn't make it as a square ...
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1answer
36 views

What are the solutions of the equation $\phi (x) =p$ with p an prime number, x an integer and $\phi $ the Euler function.

What are the solutions of the equation $\phi (x) =p$ with p an prime number, x an integer and $\phi $ the Euler function. I have actually no idea how to start with solving this problem.
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3answers
58 views

Monic polynomial $= 0 \mod p$ for all $x$

For a monic polynomial with integer coefficients (leading coefficient of $1$) $f(x)$ where $f(x) \equiv 0$ mod $p$ for all $x$, where $p$ is a prime number how do I show that the degree of the ...
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2answers
52 views

solve $3x^2 + 6x +1 \equiv 0 \pmod {19}$

I need to solve $3x^2 + 6x +1 \equiv 0 \pmod {19}$ I saw the same problem here - Solving the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$ but didn't understand how he got to the conclusion ...
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2answers
92 views

Remainder of $2^{2014^{2013}}$ when divided by $41$ [duplicate]

What is the remainder of $2^{2014^{2013}}$ when divided by $41$? The hint I have to use is that $2^{10}\equiv -1\mod 41$. Can I use the Chinese remainder theorem here? And if so, how?
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4answers
123 views

Final numbers of $2003^{2003}$ and $2003^{2003^{2003}}$

What are the three final numbers of $2003^{2003}$ and $2003^{2003^{2003}}$? Do I use the Chinese Remainder Theorem here, and if so, how?
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2answers
23 views

Prove $a,2a,\ldots,(p-1)a$ leave different remainders mod $p$

Say $p$ is a prime number and we have $a,2a,\ldots,(p-1)a$, if you then take any $ a \bmod p$ in the range of our $a$s they will all have different remainders, as long as $a$ is not $\equiv 0 ...
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2answers
498 views

Question of remainder on dividing by 7

Question : What is the remainder when $$ 10^{10} + 10^{10^2} +10^{10^3} + \ldots + 10^{10^{100}} $$ is divided by $7$?
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1answer
53 views

Finding modulo inverse if gcd is not 1

I have to find $$\frac{p^e-1}{p-1} \bmod 1000000007,$$ where $p$ is a prime number. If $\gcd(p-1,1000000007)$ is not $1$, since modular inverse of $p-1$ is not defined. Also, (p^e-1) is divisible by ...
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1answer
25 views

Calculate (A / B) mod C [closed]

someone told me: (A / B) MOD C = ((A) MOD (B * C)) / B Is that right? I have tried some testcases and it turned out to be so Please prove or give me a ...
3
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2answers
69 views

Show that there are infinitely many integer solutions to the equation $x^3+y^5=z^7$

Show that there are infinitely many integers such that $$x^3+y^5=z^7$$ and where $x^3,y^5$ and $z^7$ are all non-zero and distinct. The hint suggests to look at solutions of simultaneous equation ...
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52 views

can two different binomial coefficient be equal mod a prime?

Given two primes $p$ and $q$ with $q >p$. Can it happen $$\binom{p}{a} \equiv \binom{p}{b}\pmod q$$ for $a$ and $b$ distinct integers between $0$ and $p-1$, and $a+b$ not equal to $p$? Can anyone ...
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3answers
94 views

Chinese remainder theorem - RSA

The following is a excerpt from RSA Decryption correctness proof (section 4) : $$\begin{align} C^d &\equiv M\pmod {p} \tag{1}\\ C^d &\equiv M\pmod {q} \tag{2} \end{align}$$ Now by the ...
5
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4answers
91 views

How do I calculate $2^{47} \pmod{\! 65}$?

I'm trying to calculate $2^{47}\pmod{\! 65}$, but I don't know how... I know that: $65=5\cdot 13$ and that: $2^{47}\equiv 3 \pmod{\! 5}$ and $2^{47}\equiv 7\pmod{\! 13}$... (I used Euler) But ...
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0answers
33 views

Can I solve this mathematically? $({x^7}\mod33)^d\mod33=x$

Can I find x or d value for $({x^7}\mod33)^d\mod33=x$? Or can I make at least some assumptions of range of those values? I ask for a mathematical way because that "33" usually is a number bigger ...
5
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2answers
97 views

Solve $x^8 \equiv 3 \pmod {13}$

I need to find all solutions to $x^8 \equiv 3 \pmod {13}$. What I've tried: I know $2$ is a primitive root modulo $13$. So it is equivalent to solve $2^{8t} \equiv 2^4 \pmod {13}$ Then I get $t = ...
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1answer
24 views

Using modular arithmetic to evaluate a modulo operation

I needed to evaluate $3^{100} \pmod 7$ by hand. So, I made a list of increasing powers of $3 \pmod 7$ like so: $3^1 \equiv 3 \pmod 7$ $3^2 \equiv 2 \pmod 7$ $3^3 \equiv 6 \pmod 7$ (1) ... $3^6 ...
2
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1answer
25 views

Solving for solutions to a congruence

I am interested in solutions to this congruence: $$2^kx \equiv x \bmod m$$ Where $m$ and $x$ are known positive integers. They may not necessarily be prime or coprime. I am looking for solutions for ...
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1answer
39 views

How to perform modular

How do you solve this kind of mod when $x<n$ in $$x \bmod(n)$$ The other day I was looking at this $ 2 \bmod(7) = 2$ . How is this even possible since there will be no remainder or its always $0$ ...
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1answer
30 views

The generators of the group $\langle\mathbb{Z}_n,\oplus\rangle$ are all $g \in \mathbb{Z}_n $ for which $\gcd(g,n)=1$ [duplicate]

I'm trying to find a proof of this: The group $\langle\mathbb{Z}_n,\oplus\rangle$ is cyclic for every $n$, where $1$ is a generator. The generators of the group ...
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1answer
33 views

Clarification of a proof of Eisenstein's lemma

I'm working on a proof of quadratic reciprocity following Wikipedia's proof via Eisenstein, and one line in the proof seems unjustified: On the other hand, by the definition of $r(u)$ and the ...
2
votes
1answer
28 views

how to get maple to do linear algebra in Z_2 (integers modulo 2)

how to get maple to do linear algebra in Z_2 (integers modulo 2) I don't want it to solve and then reduce mod 2 I want it to work over Z_2 so basis([ [1,1,1], [1,-1,1 ]) = [1,1,1] If this is ...