What Are the Divisibility Patterns of Repunit Numbers?

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Divisibility patterns of repunit numbers, which are composed solely of the digit 1, reveal distinct behaviors based on the number of digits. For even-digit repunits, factors can be expressed as products involving 11 and other terms, such as (11)(101) or (11)(10101). In contrast, odd-digit repunits pose challenges, particularly when the count of digits is prime, as demonstrated by examples like 111 and 11111, which factor into products of two primes. The discussion seeks an algorithm to factor the sum of powers of ten that generate these repunit numbers. Understanding these patterns can enhance the study of number theory and divisibility.
l-1j-cho
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Hello all :)

I am studying divisibilities of
1, 11, 111, 1111, 11111 and so on
when I have even number of 1s, in other words, even number digits
obviously the numers can be factors as (11)(101) , (11)(10101), (11)(1010101)
but when I have odd number of 1s, it is pretty hard if that number is prime
for instance, 111=(3)(37), 11111=(41)(271), 1111111=(239)(4649)
can anyone give a lecture of this?

How do I find an algorithm to factor 1+10+10^2+... + 10^(2n-1)?
 
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Google for repunit numbers. You'll get lots of results.
 
Petek said:
Google for repunit numbers. You'll get lots of results.

cheers!
 
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