Questions tagged [repunit-numbers]

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

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A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder

A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder. I tried to think on it but couldn't get through. Please help.
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Prove that none of the integers $11,111,1111,…$ are squares of an integer.

Please check my proof. Thank you! Proof: $11,111,1111,...$ can all be written as follows $\underbrace{111...}_{\text{k times}}=1+10(\sum_{i=0}^{k-2}10^n)$ Let us assume $1+10(\sum_{i=0}^{k-2}10^n)=s^2$...
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Divisibility of $a_{24}$ by 7. ($a_n=\underbrace{999\cdots9 }_{n \text{ times}})$

Question: By which number is $a_{24}$ divisible by? Where $a_n=\underbrace{999\cdots9 }_{n \text{ times}}$ The solution says the answer is $7$. Here's what is given: $$a_{24}=\underbrace{999\...
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4answers
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Prove that the number $ \underbrace{11 \ldots 1}_{(p-1) \mathrm{l}^{\prime} \mathrm{s}} $ is divisible by $p$ [duplicate]

Question - Example 1.30. Let $p \geq 7$ be a prime. Prove that the number $ \underbrace{11 \ldots 1}_{(p-1) \mathrm{l}^{\prime} \mathrm{s}} $ is divisible by $p$ Proof: We have $ \underbrace{11 \...
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Pairs of Squares with repunit difference — mistake by Selfridge and Lacampagne?

Dec. 1986: C.B. Lacampagne and J.L. Selfridge: Pairs of Squares with Consecutive Digits (Math. Mag. Vol. 59, no. 5: 270 – 275) www.jstor.org/stable/2689401) contains a list of pairs of square numbers ...
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School-level problem on divisibility

I encountered the problem to show that there is an integer of the form $11111\ldots 11$ divisible by $2021$. It is easy to show that there is a number of the form $111 \ldots 11 \cdot 10^k$ ...
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1answer
53 views

Number Theory Prove a complete square 1,11,111 [closed]

I have to prove that every number in the series 11,111,1111 ... Is not a complete square, Can you give me a clue how to do this?
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2answers
69 views

repeating unit of 1's

Tried getting the last 10 digits calculated out, but couldn't figure out a pattern for the rest of more than 90 digits. Would appreciate any clue. Thanks!
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1answer
45 views

How to prove statement about octal numbers [duplicate]

How to prove for any prime $$p \neq \{2,7\}, \exists n \text{ s.t. }p \mid n$$ where n is a number that consists only of 1s in octal base. I started off but need some direction. I thought of using $$...
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34 views

Show that if N is relatively prime to 10, then there exists a multiple that consists only of 1s. [duplicate]

show that if $N$ is relatively prime to 10, then there exists a multiple that consists only of 1s. The multiple can be expressed as : $$\frac{(10^a-1)}{9}$$ thus if $\gcd(N,3)$ is not 3, using the ...
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Find the thousandth digit after the decimal point of $\sqrt N$ if N- [duplicate]

Let N be a positive integer with 1998 decimal digits, all of them 1,that is- $$N=1111......11$$ Find the thousandth digit after the decimal point of $\sqrt N$
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For a given $a$, for which values of $b$ and $c$ is the sequence $\frac{a\cdot10^n-b}{c}$ guaranteed to return integer terms with GCD=1?

One example are the repunits where $a=1$, $b=1$, and $c=9$. Another example are the numbers of the form 133...331 which are generated by the values $a=4$, $b=7$, and $c=3$. There are lots of other ...
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Prove that there's a multiple of 1997 which has only ones in its decimal expansion

A problem from an exercise book on the Pigeonhole Principle Prove that there's a multiple of 1997 which has only ones in its decimal expansion. My progress As there are infinite number of such ...
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Can $7$ be the smallest prime factor of a repunit?

Repunits are numbers whose digits are all $1$. In general, finding the full prime factorization of a repunit is nontrivial. Sequence A067063 in the OEIS gives the smallest prime factor of repunits. ...
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3answers
41 views

Suppose a = positive integer and gcd(a, 10) = 1. Prove that a divides infinitely many repunits.

Consider $S_{n} = 11...11$ where $1$ is listed $n$ times. Prove that $a$ divides $S_n$ for infinitely many values of $n$. Also consider $S_{n} = {10^{n} - 1 \over 9}$ Examples: $S_{2} = 11, S_{3} ...
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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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5answers
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What is the smallest multiple of $3^{1000}$, which has only 1’s in the digits?(Like 1, 11, 111 etc)

I think the answer is $3^{1000}$ ones, and I want to prove it with induction. $3^0 | 1$ and $1$ is the smallest. $ 3^1 | 111 $; it’s the smallest too. Let’s say it's true for $ 3^k | 111...111 $ ($...
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184 views

Find the $1000$th digit after the decimal point of $\sqrt{n},$ where $n=\underbrace{11\dots1}_{1998 \text{ 1's}}$

Find the $1000$th digit after the decimal point of $\sqrt{n}$, where $n=\underbrace{11\dots1}_{1998 \text{ 1's}}$. Obviously, $\underbrace{11\dots1}_{1998 \text{ 1's}}=\dfrac{1}{9}\left(9\cdot10^{...
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1answer
90 views

Repunit prime numbers

Consider all numbers that are written with only ones in base $10$, that is, numbers of the form $$ p_n=\sum_{i=1}^{n} 10^{i-1}=\frac{10^n-1}{9}=\underbrace{1.....1}_\text{$n$ $1$s}. $$ Here, $n$ is ...
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3answers
188 views

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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4answers
116 views

$3^2+2=11$, $33^2+22=1111$, $333^2+222=111111$, and so on.

$3^2+2=11$ $33^2+22=1111$ $333^2+222=111111$ $3333^2+2222=11111111$ $\vdots$ The pattern here is obvious, but I could not have a proof. Prove that $\underset{n\text{ }{3}\text{'s}}{\underbrace{...
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1answer
458 views

Primes with digits only 1

Let $Y(k)$ be the number consisting of $1$, repeated $k$ times. We know that $Y(2) =11$ is prime. It so happens that $Y(19)$ and $Y(23)$ are also prime. Are there any more? Regards, David
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6answers
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Is $1111111111111111111111111111111111111111111111111111111$ ($55$ $1$'s) a composite number?

This is an exercise from a sequence and series book that I am solving. I tried manipulating the number to make it easier to work with: $$111...1 = \frac{1}9(999...) = \frac{1}9(10^{55} - 1)$$ as ...
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3answers
117 views

Primes from the sum of the first n repunits $1+11+111+1111+11111+…$

Is there prime number of the form $1+11+111+1111+11111+...$. I've checked it up to first 2000 repunits, but i found none. If $R_1=1$, $R_2=1+11$, $R_3=1+11+111$, $R_n=1+11+111+...+$nth repunit. What ...
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1answer
173 views

finding the number of positive divisors for a 1111…1 1992 times

So the actual question is to prove that the number of positive divisors is even. But to do that I have to find the number of positive divisors for 111.....1(1992 1's). I know that I should try to find ...
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2answers
811 views

Is $\underbrace{11111\ldots1}_{91\text{ times}}$ a prime number? [duplicate]

Prove that $$\underbrace{11111\ldots1}_{91\text{ times}}$$ is a composite number and not a prime. Please give full steps of proving. I tried and found that it is divisible by $1111111$ and $...
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1answer
228 views

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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3answers
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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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1answer
152 views

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 ...
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3answers
360 views

prove that there are infinitely many numbers of the form $x = 111…1$ such that $31\mid x$

I need to prove that there are infinitely many numbers of the form $x = 111....1$ such that $31\mid x$ what I tried - I wrote x as $\sum_0^{n-1} 10^i$ I know that $(10,31) = 1 $ now I'm stuck .. ...
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What is $\underbrace{555\cdots555}_{1000\ \text{times}} \ \text{mod} \ 7$ without a calculator

It can be calculated that $\frac{555555}{7} = 79365$. What is the remainder of the number $5555\dots5555$ with a thousand $5$'s, when divided by $7$? I did the following: $$\begin{array} & 5 \ ...
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5answers
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Primes dividing $11, 111, 1111, …$ [duplicate]

How can I prove that every prime except 2 and 5 divide infinitely many of the following integers $11, 111, 1111, ...$ ?
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1answer
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Repunit Divisibility

We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$. Lemma: Let $n$ be a positive integer and $GCD(10,n) = 1$. Then, there exists a $k$ such that $R(k) \equiv 0$ $ $ $ $...
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Why do repunit primes have only a prime number of consecutive $1$s?

Repunit primes are primes of the form $\frac{10^n - 1}{9} = 1111\dots11 \space (n-1 \space ones)$. Each repunit prime is denoted by $R_i$, where $i$ is the number of consecutive $1$s it has. So far, ...
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3answers
5k views

When is the number 11111…1 a prime number?

For which $n$ is the sum: $$\sum_{k=0}^{n}10^k$$ a prime number? Are they finite?
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10answers
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Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer

Please help me with solving this : prove that none of $\{11, 111, 1111 \ldots \}$ is the square of any $x\in\mathbb{Z}$ (that is, there is no $x\in\mathbb{Z}$ such that $x^2\in\{11, 111, 1111, \ldots\}...
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5answers
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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 ...