# Questions tagged [repunit-numbers]

For questions about repunit numbers, that is, numbers that contain only the digit 1.

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### For any fixed integer $a \gt 1$, how do you prove that $\frac{a^p-1}{a-1}$ is not always prime given prime $p \not \mid a-1$?

I assumed this would be easy to prove but it turned out to be quite hard since the go to methods don't work on this problem. Once we fix any $a\gt 1$, we need an algorithm to produce a prime $p$ that ...
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### $n$- relatively prime with $10$, then show that there exists another natural number $m$ such that all its digits are $1'$s and $m$ is divisible by $n$ [duplicate]

If there is a natural number $n$ relatively prime with $10$, then show that there exists another natural number $m$ such that all its digits are $1'$s and $m$ is divisible by $n$. Approach: Let the ...
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### Prime Repunit Numbers [duplicate]

proof that if a repunit number is prime n has to be prime So a repunit number is a number that it's all digits are 1. For example $R_{2} = 11$ $R_{7} = 1111111$ and so on. Repunit numbers can be ...
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### A similar (little) Fermat's Theorem result and repunit multiple

I got this exercise in arithmetic class (I'm a french student but let me translate the problem) In this thread I only talk about questions from question 2) on the paper. Let n and p be 2 integers ...
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### Let $a_n = 1 . . . 1$ with $3^n$ digits. Prove that $a_n$ is divisible by $3a_{n−1}$. [closed]

Let $a_n = 1 . . . 1$ with $3^n$ digits. Prove that $a_n$ is divisible by $3a_{n−1}$. Is there any way to solve this question without mathematical induction?
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### Show that the integers 1111, 111111, 11111111, … (numbers formed by an even number of numbers 1) are all composed. [duplicate]

I am studying congruence and I am trying to solve this problem, but I cannot think of a way to do this. Would anyone be able to help me? Show that the integers 1111, 111111, 11111111, ... (numbers ...
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### A repunit is a number that contains only “ones” (for example $111$, $1111111$,….). Prove that one can find a repunit divisible by $1973$

It is a pigeonhole problem. I have already known that there are $1972$ remainders in total and the two numbers which have the same remainder can be subtracted and the difference between the two ...
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### Repunit prime numbers

Consider all numbers that are written with only ones in base $10$, that is, numbers of the form $$p_n=\sum_{i=1}^{n} 10^{i-1}=\frac{10^n-1}{9}=\underbrace{1.....1}_\text{n 1s}.$$ Here, $n$ is ...
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### How to prove 1111…11 (91 digits) is a prime or composite number? [duplicate]

How to prove $1111......11$ ($91$ digits) is a prime or composite number? My Approach: $1111......11$ can be expressed as $10^{0}+10^{1}+10^{2}+...…..+10^{90}$ Using summation of a geometric ...
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### Prove that every prime except for $2$ and $5$ is a factor of some number of the form $111\dots1$ [duplicate]

How do we prove that for every prime $p\neq2,5$ there exists a positive integer $n$ such that $(10^n-1)/9$ is divisible by $p$? BTW, I suspect that it holds not only in base $10$, but also in every ...
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### The sequence of integers $1, 11, 111, 1111, \ldots$ have two elements whose difference is divisible by $2017$

Prove that the sequence $\{1, 11, 111, 1111, .\ldots\}$ will contain two numbers whose difference is a multiple of $2017$. I have been computing some of the immediate multiples of $2017$ to see how ...
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### Prime and Repunits

Prove that: For any integer $n > 5$, if $n$ divides $\dfrac{10^{n-1}-1}{9}$, then $n$ is a prime number. This can also be generalized further as If $n$ is an integer > 5 and divides a ...
### Prove that every number ending in a $3$ has a multiple which consists only of ones.
Prove that every number ending in a $3$ has a multiple which consists only of ones. Eg. $3$ has $111$, $13$ has $111111$. Also, is their any direct way (without repetitive multiplication and ...