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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$p>3$ 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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59 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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70 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
70 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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73 views

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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81 views

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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31 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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114 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
28 views

Congruents and modulo question- counterexample

I have tired an couple and none seem to be working
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72 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. I like thinking about this problem, it is ...
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24 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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34 views

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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47 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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39 views

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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45 views

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
15 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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26 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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256 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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1answer
36 views

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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58 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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43 views

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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33 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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63 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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37 views

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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Quadratic function as permutation of sequence

Say I have a $n \in \mathbb{N}$ and $$a_i := (1,2,...,2^n)$$ and two function $$f(i) = \sum_1^i i = \frac{i(i+1)}{2}$$ $$g(i) = f(i) mod 2^n$$ When I now look at a new sequence $$b_i = (a_{g(0)}, ...
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324 views

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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18 views

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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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
23 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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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. $$\frac{n^2}{x}-\left\lfloor\frac{n^2}{x}\right\rfloor+\frac{2n+1}{x}-\left\lfloor\frac{2n+1}{x}\right\rfloor>1$$ ...
3
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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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24 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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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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33 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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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 $7$, ...
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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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Number of solutions to a congruence in a PID

This is the proof that there are $(a,m)$ solutions to $ax\equiv b \mod m$ for the ring $\mathbb{Z}/m\mathbb{Z}$. Where does this proof not hold for a general PID, allowing for (for instance) an ...
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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 ...
3
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346 views

Frog Jump Problem

Lotus leaves are arranged around a circle. A Frog starts jumping from one leaf in the manner described below. In the first jump it skips one leaf,next jump it skips two,three the next jump ...
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Show that there exists no integer coordinates on curve

Problem: Show that there does not exist any integer coordinates to the curve $$y = \frac{x^2-3}{4}, x\in \mathbb{R}.$$ My attempt: The problem is equivalent of saying that there does not exist any ...
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Property of modulo congruation

If I have: $$a^b \equiv 1 \mod xy$$ where $x,y$ are primes, is then true that: $$ a^b \equiv 1 \mod x$$ $$ a^b \equiv 1 \mod y$$ I don't sure if this is true, because I don't know how can I prove it ...
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40 views

The addition table for $\mathbb Z/4$ - modular arithmetic

"Write down the addition table for $\mathbb Z/4$ " Could someone please give one or two hints? And what does them mean with $\mathbb Z/4$? They have never used that notation before. Do them just mean ...
3
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54 views

$\operatorname{lcm}(n,m,p)\times \gcd(m,n) \times \gcd(n,p) \times \gcd(n,p)= nmp \times \gcd(n,m,p)$, solve for $n,m,p$?

$\newcommand{\lcm}{\operatorname{lcm}}$ I saw this in the first Moscow Olympiad of Mathematics (1935), the equation was : $$\lcm(n,m,p)\times \gcd(m,n) \times \gcd(n,p)^2 = nmp \times \gcd(n,m,p)$$ ...
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18 views

unique solution for mod equation

If I have these equations $a\equiv b \pmod c$ $d\equiv e \pmod c$ All known except $c$ and $\gcd(b−a,e−d)=1$ how do I find the unique solution for $c$? and if the gcd!= 1 how do I find some ...
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27 views

Help for solving quadratic residue.

I am solving CRYPTO1 problem. This problem requires to solve following equation: $$x^2=q(\mod p)$$ where I am given $p=4,000,000,007$ and q is also given. I followed ...
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1answer
44 views

question about the forms of prime numbers

I was thinking about primes earlier and I thought of a hypothesis that I have been unable to prove. I was wondering whether it was a known theorem and whether anyone knows a proof or can prove (or ...
0
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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 ...
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2answers
64 views

Can we say that $a=b \implies a \equiv b \pmod{0}$?

I'm wondering if we can write $a=b$ as $a \equiv b \pmod{0}$. Because the last congruence satisfies $b-a=0\times k$ $\implies b-a=0$. Which is really $b=a$.