# 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...