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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If $a, b$ are relatively prime proof.

Prove that if $a$ and $b$ are relatively prime integers and $ab$ is a perfect square so are $a$ and $b$. Show by counterexample that the relatively prime condition is necessary. I dont know how ...
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Given any integers $a,b,c$ and any prime $p$ not a divisor of $ab$, prove that $ax^2+by^2\equiv c\pmod{p}$ is always solvable.

The fact that there are $\dfrac{p+1}{2}$ quadratic residues seem to me to help solving the question, but I don't know how to go on from that point. Could you give me any hint?
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52 views

Divisibility of sum of exponents

Consider the sequence $$r, \ ra, \ ra^2, \ ra^3, ... \ , ra^n \mod M $$ such that: $$ ra^{n+1} \equiv r \mod M$$ and $a \ne 1$ and $a,r$ are both coprime to $M$ Is it always true then that: ...
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62 views

Let $p$ be a prime number such that $p \equiv 3( mod$ $4$). Show that $x^2$ $\equiv$ -1 $(mod$ $p)$ has no solutions. [duplicate]

Let $p$ be a prime number such that $p \equiv 3( mod$ $4$). Show that $x^2$ $\equiv$ -1 $(mod$ $p)$ has no solutions. I noticed that this is equivalent to proving $ x^2\equiv 2(2k+1) $ $(mod $ $p)$. ...
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1answer
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Modulus Implication

If $a, \ b, \ c, \ d$ are integers and $a \equiv b \pmod c$ then $d^a \equiv d^b \pmod c$. True or false? I changed this statement to If $a,b,c,d$ are integers and $c\mid (a-b)$ then $c\mid (d^a - ...
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2answers
125 views

Prove that $a^n+b^n \equiv (a+b)^n \mod n$, if $n$ is prime and $a,b$ are integers.

What is the best method to prove that if $n$ is prime and $a,b$ are integers $a^n+b^n \equiv (a+b)^n \mod n$, ?
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1answer
61 views

Helping solve mod problems

I am having trouble solving the below problems. My teacher taught us to write out the solutions by hand.. but I really think there is an easier way to do the higher numbers. Thanks!
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1answer
58 views

Solution for generalized Euler's Theorem $a^m\equiv a^{m-\phi(m)} \pmod{m}$?

The above identity holds for any integer $a$. Since my solution(?) does seem neither elegant nor rigorous enough, I want to get some advice to improve it. My solution: If $(a,m)=1$, this identity is ...
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1answer
64 views

How do I find the inverse of $e \bmod (p-1)(q-1)$?

I'm trying to find this inverse modulo to set up a solution for an RSA cipher. I haven't the slightest how to go about this. When I looked up the formula for such a question, it states: $$ d \equiv ...
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32 views

Divisibility Method

Is there exist any known method to find divisibility rule of each and every rational number in any numeral system by analysing its reciprocal. And additionally it will give the remainder on division ...
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2answers
61 views

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers. Progress: $F_{11}=89$ . I believe you should find the period of $F_n \bmod 89$ and use that to solve it. But I'm not not ...
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1answer
62 views

Quadratic residue modulo $p$ iff quadratic residue module $p^k$

Let $p$ be an odd prime, $a\in \mathbb{Z}$ with $(a,p)=1$. I am trying to show that if $a$ is a square modulo $p$ then it is a square modulo $p^k$. I managed to prove this using an exponential ...
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1answer
38 views

Large Modular Arithematic Exponentiation [duplicate]

How do I calculate $2^{65536} \pmod{2^{31} -1}$ $3^{256} \pmod{2^8 +1}$ Please help?
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41 views

$p>3$ prime, show that there exists $0<x,y<\sqrt{p}$ so that $p$ divides $cx-y$

Let $p>3$ be a prime number that does not divide $c$. Show that if $p>3$ there exists $x$ and $y$ with $0<x,y<\sqrt{p}$, such that p divides $cx-y$. I believe I've shown the above but for ...
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4answers
64 views

$a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$

Given that $a \equiv b \pmod n$ and $c\equiv d \pmod n$, I need to prove that $ac \equiv bd \pmod n$ So far, I've only managed to deduce that $a+b \equiv c+d \pmod n$. I don't know if this is usable, ...
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1answer
84 views

Find the largest integer $n$ such that $10^n$ divides $10^6!$

Let $N=10^6!$ Find the largest integer $n$ such that $10^n$ divides $N$. Furthermore, compute the first digit and the last non-zero digit of $N$. I have some ideas that you should be able to use ...
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1answer
76 views

Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers. I have tried using the fact that $F_{n^k} \bmod F_n = 0, k=1,2,3,...$ but that doesn't get me anywhere. Thanks!
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2answers
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Solving $x^2 + 96=0$ in $\mathbb{Z}_{100}$

I'm trying to find all solutions to $x^2 + 96=0$ in $\mathbb{Z}_{100}$. $x^2 + 96 \equiv 0 \bmod 100$ implies that $x^2 + 96 \equiv 0 \bmod 2$ and $x^2 + 96 \equiv 0 \bmod 5$. $$x^2 + 96 \equiv 0 ...
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3answers
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Does $p \equiv q \pmod a \implies p \bmod a = q \bmod a$?

I'm trying to understand the notation $p \equiv q \pmod a$. Does does it implies that $p \bmod a = q \bmod a$? for example: $$ \begin{align} 5 \bmod 7 &= 5 \\ 12 ...
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1answer
42 views

Group of numbers in residue number system of first $n$ primes

Denote $$\Pi_n = \Pi_{i=1}^n p_i$$ ie. the product of the first $n$ primes. Assume we have a number $m$. Denote $$L(m) = k \iff \Pi_{k-1} < m \le \Pi_{k}.$$ Assume also $L(1)=1.$ Now, if $L(m) = ...
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177 views

What is the remainder when $24^{1202}$ is divided by $1446$?

I tried remainder theorem but that does not simplify it. I tried factorizing $1446$ as $2\cdot3\cdot241$ and got remainders when numerator is divided by $2,3$ and $241$ individually but then I did ...
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1answer
89 views

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$. This is a problem from a selection to IMO 2014.
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28 views

Why does $y^{pq} ≡ y $[mod $pq$] imply $y^{pq} ≡ y [$mod $p$] and $y^{pq} ≡ y [$mod $q$]? $p, q$ prime.

Why does $y^{pq} ≡ y $[mod $pq$] imply $y^{pq} ≡ y [$mod $p$] and $y^{pq} ≡ y [$mod $q$]? where $p, q$ prime. I can't see it from re-writing it as $y^{pq} = y + kpq$ for some integer $k$, as you ...
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Solve by using modulo. Or something else

An infinite sequence of positive integers $a_1, a_2,\ldots$ has the properties that for $k\geq2$, the $k^\text{th}$ element is equal to $k$ plus the product of the first $k-1$ elements of the ...
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2answers
54 views

Property of modulo division

I wanted to check if it is true, that $$a^{3b} \pmod n = (a^{b} \pmod n)^{3}\ ?$$ For example when $a = 2, b = 4, n = 5$ I have that $2^{12} \mod 5 = 1$ and $(2^4 \mod 5)^3 = 1$ Is that always true, ...
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Property of Modular arithmetic

If I know that $$g^a \neq 1 \mod b$$ is that always true that if I will take a positive integer $c$ and count $(g^a)^c$, then $$(g^a)^c \neq 1 \mod b$$?
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Why is $a^{n/2} \equiv -1 \mod p$ but not necessarily -1 modulo a composite?

I'm going over a review sheet in preparation for my number theory final. We are asked to prove the following: |a| = 2r, show that $a^r \equiv -1\mod p$ a prime. Does this hold modulo n, where n is a ...
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Cannot find length of repeating block in decimal expansion for $\frac{17}{78}$

I am trying to find the length of of the repeating block of digits in the decimal expansion of $\frac{17}{78}$. On similar problems, that has not been an issue. Take for instance $\frac{17}{380}$. My ...
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1answer
17 views

Using information from a congruence to factor a number

I am being asked to factor $15347$ given that $7331^2 \equiv 1460^2 \pmod{15347}$. I've tried playing around with each of the numbers -- prime factorization, gcd, lcm, etc., but I can't find a ...
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1answer
30 views

finding $m$ from $c = m^e \pmod{n}$

I'm working through an RSA encryption example, and I'm being asked to solve $c = m^e \pmod{n}$ for $m$ given c, e, and n (along with its factorization.). Since I already have that information ...
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4answers
264 views

Find number x such that $x\equiv 4^{1002}\pmod{55}$

Find a natural number x, for $0 \le x \le 54$ such that is a solution for the following equation: $$x\equiv 4^{1002}\pmod{55}$$ This question was asked in an exam, so I expect that the answer is ...
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Simple modulo congruence operation

Find a number $n$, with $0 \leq n < 15$, so that $6 \times 7$ is congruent to $n$ modulo $15$ Please explain your work.
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Showing equality of primitive roots with quadratic non-residues.

Suppose that $p$ and $q = 2p + 1$ are both odd primes. Show that the $p − 1$ primitive roots of $q$ are precisely the quadratic non-residues of $q$, other than the quadratic non-residue $2p$ of ...
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63 views

Matrix Induction proofs using $\rm mod$.

Please could someone advise me on where to even start with this question. Thanks in advance Let $A$ be an $n\times n$ matrix, for some $n\geqslant 1$. Given any prime number $p\gt1$, write $A_p$ for ...
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4answers
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How to find all elements in Z/80 that have multiplicative inverses.

I need to find all the elements in Z/80 that have multiplicative inverses. Z/80 is not a field, so I know not every element will have an inverse. Is there a shorter way than just writing out the ...
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54 views

How to calculate sum of digit of a power

Find the sum of the digits of: $$\left\lfloor\frac{k^{h+1}-1}{h-1}\right\rfloor$$ I need to calculate sum of digits in answer. Note as $k$ and $h$ can be a very big value, answer is getting ...
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3answers
66 views

Calculating $x^{103} \equiv 2 \pmod{143}$

I need to find x, given that: $0\leq x \leq 143$ and $x^{103}\equiv 2 \pmod{143}$. I tried to use Euler's theorem $p(143)=120$, but it didn't help.
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Sum of the jumbled digits of $abc_{10}$ is $3194$

In the book that I am reading, the author denotes $abc_{10}$ as $100a+10b+c$ where $a, b, c \lt 10$. So if $a = 3$, $b=2$ and $c=8$ then $abc_{10} = 328$. The author asks the following problem: In ...
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What are primitive roots modulo n?

I'm trying to understand what primitive roots are for a given $\bmod\ n$. Wolfram's definition is as follows: A primitive root of a prime $p$ is an integer $g$ such that $g\ (\bmod\ p)$ has ...
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1answer
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Matrix double modulo multiplication to get identity

I have to multiply to matrices A and B which can consist of numbers 0,2,3,4,5,6 to get an identity matrix, however multiplication happens with moduli after every ...
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1answer
36 views

How to show $2^{k+2}$ divides $3^{2^k}-1$ but $2^{k+3}$ doesn't?

I've got a task: Find highest power of 2 that divides $3^{2^k}-1$ so i wrote few terms and guessed that it's $2^{k+2}$, now i should show it. I tried by induction, but what i got appeals to me as a ...
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1answer
29 views

How to prove $N=13\times12v+6\times19u$ is a solution for the system?

Well, I have a system of congruences it is : $$n\equiv13\pmod{19}$$ $$n\equiv6\pmod{12}$$ I'm trying to prove that for any pair of integers $(u,v)$ the number $N=13\times12v+6\times19u$ is a solution ...
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37 views

Floor function inequality: $\frac{n^2}{x}-\left\lfloor\frac{n^2}{x}\right\rfloor+\frac{2n+1}{x}-\left\lfloor\frac{2n+1}{x}\right\rfloor<1$

I would like to dissect the following inequality to figure out its properties. $$\frac{n^2}{x}-\left\lfloor\frac{n^2}{x}\right\rfloor+\frac{2n+1}{x}-\left\lfloor\frac{2n+1}{x}\right\rfloor>1$$ ...
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1answer
63 views

How to show $(n-1)^3n^3(n+1)^3$ is divisible by 7 and 9?

Yeah it looks like a basic, really elementary question, but i'm having hard time with it. First i tried to show that it's divisible by 9 $$(n-1)^3n^3(n+1)^3 = ((n+1)(n-1))^3n^3 = (n^2-1)^3n^3 = ...
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1answer
27 views

Find all values of parameter A such that two system of congruences are equal

I'm starting to learn some elementary number theory and i came across a task i don't know how to solve. $$x \equiv 5 (mod \ 6)$$ $$x \equiv A (mod \ 35)$$ and the second one $$x \equiv A (mod \ ...
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86 views

Discrete log modulo prime

I'm trying to understand properties of the discrete logarithm problem modulo a prime. For a prime $p$, an $\alpha \in \mathbb{Z}_p^*$ and $a \in \mathbb{Z}_{p-1}$ why does $\alpha^x \equiv 1 \mod p$ ...
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0answers
136 views

solving congruence equation system modulo prime

I need to solve a congruence system like this: $30f_0+26f_1+8f_2+38f_3+2f_4+40f_5+20f_6 \equiv 0 \pmod{41}$ $38f_0+2f_1+40f_2+20f_3+30f_4+26f_5+8f_6 \equiv 0 \pmod{41}$ ...
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4answers
71 views

Find the smallest integer $n\geq1$ such that $35n$ is a perfect square and $n/7$ is a pefect cube.

Find the smallest integer $n\geq1$ such that $35n$ is a perfect square and $n/7$ is a pefect cube. What I have so far: we express the prime factorizations of $35$ and $7$ as $5\cdot7$ and $7$, ...
2
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0answers
49 views

System of equations - modular arithmetic

I am asked to solve the following..... Let $n\in \mathbb{N}$ and suppose that $a,b,c,d,k,l\in\mathbb{Z}$. Consider the system $ax + by \equiv k$ mod $n$ and $cx+dy \equiv l$ mod $n$. Let $D=ad-bc$. ...
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2answers
30 views

In modular arithmetic is $m^{1/x}$ =! $(m^x)^{-1}$

In modular arithmetic is $m^{1/x}$ =! $(m^x)^{-1}$ as we always use inverse instead of reverse in multiplicative group.why reverse operation is not used in modular arithmetic and if one want to use ...