2
votes
3answers
80 views

remainder when 67896789…(300 digits) divided by 999

What is the remainder when 678967896789... (300 digits)is divided by 999? i tried to divide it manually to find some pattern in remainder. But was getting bit lengthy. so please suggest me some short ...
3
votes
1answer
77 views

Remainder on dividing $10^{n} + 10^{n-1} + … + 10^{1} + 10^{0}$ by x

Given a positive integer $n$, consider the number $y=10^{n}+10^{n-1}+$$...+ 10^{1}+10^{0}$. I need to find the remainder when $y$ is divided by a natural number $x$. e.g. $111111$ $\%$ $2123$ = ...
0
votes
1answer
42 views

Under what conditions can we obtain $a \equiv 1 \pmod{mn}$ from $a \equiv b \pmod{m}$ and $b \equiv 1 \pmod{mn}$?

If $a \equiv b \pmod{m}$ and $b \equiv 1 \pmod{mn}$, are there any conditions under which we can conclude that $a \equiv 1 \pmod{mn}$? Here $m$ and $n$ are any integers; $a$ and $b$ are both coprime ...
1
vote
1answer
46 views

Residue class of a huge repunit modulus a huge number

Given a number with only 1: X = 1111...1 (N times 1 in total), and another number M, I want ...
0
votes
2answers
92 views

Modulus of large powers

Given an array of N integers where $2 ≤ N ≤ 2×10^5$ and each element in array is less than $10^{16}$. Now I am given a variable $X$ that can also go up to $10^{16}$. We need to find if $X \mid ...
0
votes
2answers
79 views

Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$

Let $a,b,c$ be co-prime integers $>2$ . Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$.
3
votes
0answers
101 views
+50

Prove modular equivalence related to carmichael function.

Statement if $n=p_1p_2p_3\cdots p_r$ (prime numbers are distinct) and $2 \le a \le n-1$ then prove that $$a^{k\lambda} \equiv \left(1+\sum_{p_i\mid\gcd(a,n)}\left(\dfrac{n}{p_i} \right) ...
0
votes
1answer
29 views

What is the smallest number of times that the digit 1 can appear in N?

All the digits of the positive number $N$ are either $0$ or $1$. The remainder after dividing $N$ by $37$ is $18$. What is the smallest number of times that the digit $1$ can appear in $N$? I have ...
4
votes
2answers
103 views

Sum of two squares modulo p

I have heard somewhere that for all primes $p$, for all $k$, there exist $x, y$ s.t. $x^2 + y^2\equiv k \pmod{p}$? I recall that the proof is very elementary, but I can't remember such a proof. How ...
1
vote
1answer
35 views

Digit wise modulo for calculating power function for very very large positive integers

I am writing a code to calculate $P^Q$ where $P$, $Q$ are positive integers which can have number of digits up to $100000$. I want the result as $r = P^Q \pmod{10^9+7}$, where $10^9+7$ is a prime ...
1
vote
4answers
149 views

Why didn't Fermat provide proofs of his theorems?

Apparently Fermat stated but didn't provide proofs of various theorems named after him, including Fermat's little theorem, Fermat's theorem on sums of two squares, Fertmat's polygonal number theorem, ...
2
votes
0answers
92 views

“Remainder” operation in mod 2^32

I debated posting this here, in the cryptography SE, or the programming SE. Obviously, I chose here, but I'm not confident in my choice... I'm attempting to "undo" a function, but I've hit a slight ...
44
votes
4answers
3k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
1
vote
3answers
67 views

Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that ...
0
votes
1answer
193 views

Why elements of the set can be Goldbach pairs for a given even number?

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
0
votes
1answer
36 views

Reverse Modulus Operator with given condition

I have an equation: $$ x^2 \mod p = z $$ $p$ and $z$ are given. $x$, $p$ and $z$ are positive integers and a maximal value of $x$ is given (say $M$). $p$ is a prime. How can i calculate (multiple ...
1
vote
0answers
12 views

Jacobi Form and its Fourier expansion

Let k,m be non negative integers. A Jacobi form of weight k and index m is a holomorphic function f on $\mathbb{H} x \mathbb{C}$ (where $\mathbb{H}$ denotes the upper half plane) satisfying the ...
1
vote
3answers
36 views

What are the members of the set $A=7^{n}+5^{n}(mod35)$

I have this set $A=${$\ x \ \in \mathbb{N}|\ \exists \ n \ \in \mathbb{N}:$ $x \equiv 7^{n}+5^{n}$ (mod $35$) $ $, $ 35\gt x\ge 0$} I want to know how many members has this set? thanks in advance
0
votes
2answers
36 views

How to prove that $2^{2^{m}}-1 \equiv 0$ (mod $2^{2^{n}}+1$)

I'd like to solve this problem but I can't $\exists \ m,n \ \in \mathbb{Z}$ & $ m\gt n\ge 0$ $2^{2^{m}}-1 \equiv 0$ (mod $2^{2^{n}}+1$) Any ideas? Thanks in advance.
0
votes
1answer
41 views

Polynomial Diophantine Equations

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$ Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , ...
1
vote
2answers
79 views

Solving $a^2+b^2\equiv 0$ mod $c$ for distinct integers $a,b,c$ constrained between two consecutive squares

Show that for any natural number $n$, between $n^2$ and $(n+1)^2$ one can find three distinct natural numbers $a,b,c$ such that $a^2+b^2$ is divisible by $c$. A friend and I found a general case that ...
0
votes
1answer
28 views

Proof that the repeating block of digits in 1/x is at max x-1?

The question is self-explanatory, I suppose. Example, the maximum number of digits in the repeating block of 1/17 is 16. Thanks in advance.
2
votes
1answer
45 views

The existence of primitive algebraic units modulo 3

Consider the problem of computing $$\sqrt{2} \mod 3 $$ Whereas we seek a number $n$ such that $n^2 \equiv 2 \mod 3$ and furthermore it is known that both $n$ and $2n$ will satisfy this property, ...
1
vote
1answer
34 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: ...
0
votes
1answer
99 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 ...
2
votes
2answers
31 views

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 ...
0
votes
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.
1
vote
3answers
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.
2
votes
0answers
22 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 $D=ad-bc$. ...
0
votes
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 ...
-4
votes
3answers
150 views

Congruence $x^n\equiv2 \pmod{13}$ (Multiple Choice) [closed]

I was trying to solve the following problem.Please help. Consider the $x^n\equiv2 \pmod{13}$. It has a solution for $x$ if $n=5$ $n=6$ $n=7$ $n=8$ It may have more than one correct options. Thnx ...
3
votes
3answers
61 views

Solve for x in $5 \equiv 128x$ (mod 59)

I'm just running a blank here in review for finals and cannot seem to figure it out for the life of me. I want to say that you must use extended euclidean algorithm somehow, and I checked that the ...
0
votes
3answers
32 views

Modular equations, find x

Problem: Find an integer $x$ such that $x = 5\pmod 8, x = 3 \pmod 9, x = 4 \pmod 7$. Attempt: By the Chinese Remainder Theorem " Suppose $a_1,a_2,...a_k$ are integers pairwise relatively prime ...
0
votes
1answer
43 views

For every positive integers $a$ and $n$, is it true that: $a^{n−1} \equiv 1 \pmod n \iff \gcd(a,n) = 1$?

Based on Fermat's little theorem or on Euler's theorem, is the statement $$a^{n−1} \equiv 1 \pmod n \iff \gcd(a,n) = 1$$ true for every positive integers $a$ and $n$?
2
votes
1answer
34 views

Euclidean Algorithm for Modular Inverse, with negative numbers

I might be on to something quite simple which I'm failing to see, while calculating modular inverses. For example, calculating 7x = 5 (mod 12) Which is the same as saying: 7x - 5 = 12k Which ...
6
votes
1answer
152 views

Congruences and prime powers

I have just a small question that probably is not hard to answer, but I could not find and elegant solution to this question. Let $p$ and $q$ be prime numbers. $$5^q\equiv 2^q \pmod p$$ $$5^p\equiv ...
0
votes
3answers
36 views

Modular Arithmetic

I am doing some exam preparation and can't figure out how to do the following question. It seems to be a regular question and was wondering if anyone who could tell me an approach to this style of ...
1
vote
2answers
84 views

Last Two Digits Problem

I'm trying to find the last two digits of ${2012}^{2012}$. I know you can use (mod 100) to find them, but I'm not quite sure how to apply this. Can someone please explain it?
0
votes
1answer
31 views

Question regarding solving a modulo equality

Two Equations: ab % c = d (ci + d) % c = d, i $\in \mathbb N$ I want to solve for b given the above two equations with a, c, and d known. ab = ci + d b = (ci + d) / a i = (k + an), n $\in ...
1
vote
8answers
163 views

Calculate the last digit of $3^{347}$

I think i know how to solve it but is that the best way? Is there a better way (using number theory). What i do is: knowing that 1st power last digit: 3 2nd power last digit: 9 3rd power last digit: ...
1
vote
1answer
47 views

Integer solutions of the equation $a^{n+1}-(a+1)^n = 2001 $

I am doing number theory and I came across that question $a^{n+1}-(a+1)^n = 2001 $. Find the integer solutions and show that they are the only solution. I really tried hard but i am nowhere near ...
1
vote
1answer
37 views

Proof on Divisibility of Binomial Coefficients

Prove that $\exists \ i$ $(0 \lt i \lt n)$ such that $$ n \nmid {n \choose i} $$ $\forall \ n$ such that $n \gt 0$ and $n$ is a composite Number.
1
vote
4answers
52 views

Problem modulo $p$.

Let $p$ be a odd prime, prove that $1^p+2^p+...+(p-1)^p \equiv 0 \mod p$ I'm not sure how to do this, thanks.
2
votes
2answers
54 views

Describe all integers a for which the following system of congruences (with one unknown x) has integer solutions:

$$x\equiv a \pmod {100}$$ $$x\equiv a^2 \pmod {35}$$ $$x\equiv 3a-2 \pmod {49}$$ I'm trying to solve this system of congruences, but I'm only familiar with a method for solving when the mods are ...
2
votes
1answer
44 views

Problem on Number of Quadratic Residues

We have two primes $p,q$ and an integer $a$ such that $$\gcd(a,pq)=1$$ How to prove that for the following congruence $$x^2 \equiv a \mod pq$$ either there will be $4$ solutions or $zero$ solutions. ...
1
vote
4answers
42 views

How to find $k$ in $6^{k} \equiv 2 \mod {13}$

Find for which $k$ is $6^{k} \equiv 2 \mod {13}$ I'm having trouble with these types of question in my cryptography class. This is part of Diffie–Hellman algorithm for calculating a shared key. ...
1
vote
2answers
80 views

Determine the last two digits of $3^{3^{100}}$

Determine the last two digits of $3^{3^{100}}$ This is one of the problems in the past exam my modern algebra course. I think I need to use euler-fermat theorem but can't figure out how to use it for ...
1
vote
4answers
78 views

Proving congruence modulo, number theory

The task is to prove $24^{31}\equiv 23^{32}\pmod {19}$. I'm trying to use Fermat's little Theorem and so far I only found that $24^{31}\equiv 19\pmod{19}$. Would proving that $17\mid23^{32}$ prove ...
1
vote
3answers
442 views

Find the smallest positive integer x such that 2015! ≡ x (mod 2017)

Q. The next year that is a prime is 2017. Find the smallest positive integer x such that 2015! ≡ x (mod 2017). So, this is what I have; By Wilson’s theorem, (2017-1)! ≡ -1 (mod 2017) ⇒ 2016! ≡ -1 ...
1
vote
1answer
181 views

Modular exponentiation with Chinese Remainder Theorem

I'm learning modular exponentiation with Chinese remainder theorem. I found a great answer from below ...