Decoding Repunit Numbers: Exploring the Divisibilities of 1, 11, 111, and Beyond

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In summary, the divisibilities of 1, 11, and 111 are as follows: 1 is divisible by all numbers, 11 is divisible by 1 and itself, and 111 is divisible by 1, 3, 37, and itself. To determine if a number is divisible by 1, 11, or 111, we can use the divisibility rules, such as the difference between the sum of its even and odd digits. Any number can be divisible by 1, 11, or 111, and there are other numbers besides those listed that are divisible by these three numbers. The divisibility of 1, 11, and 111 can be useful in math and science,
  • #1
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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  • #2
Google for repunit numbers. You'll get lots of results.
 
  • #3
Petek said:
Google for repunit numbers. You'll get lots of results.

cheers!
 

1. What are the divisibilities of 1, 11, and 111?

The divisibilities of 1, 11, and 111 are as follows:

  • 1 is divisible by all numbers.
  • 11 is divisible by 1 and itself.
  • 111 is divisible by 1, 3, 37, and itself.

2. How can we determine if a number is divisible by 1, 11, or 111?

To determine if a number is divisible by 1, 11, or 111, we can use the divisibility rules:

  • A number is divisible by 1 if it is any number.
  • A number is divisible by 11 if the difference between the sum of its even digits and the sum of its odd digits is either 0 or a multiple of 11.
  • A number is divisible by 111 if the difference between the sum of its hundreds, tens, and ones digits and the sum of its thousands, ten thousands, and hundred thousands digits is either 0 or a multiple of 111.

3. Can any number be divisible by 1, 11, or 111?

Yes, any number can be divisible by 1, 11, or 111. This is because the definition of divisibility for each of these numbers is not limited to specific numbers, but rather applies to all numbers.

4. Are there any other numbers that are divisible by 1, 11, or 111?

Yes, there are other numbers that are divisible by 1, 11, or 111. In addition to the numbers listed in the first question, there are also infinite numbers that can be divisible by these three numbers. For example, 222 is divisible by 1, 11, and 111.

5. How can the divisibility of 1, 11, and 111 be useful in math or science?

The divisibility of 1, 11, and 111 can be useful in various ways in math and science. For example, divisibility rules can be used to quickly determine if a number is divisible by 1, 11, or 111, which can be helpful in mental calculations. In science, divisibility can also be important in analyzing patterns and relationships between numbers, which can lead to further understanding and discoveries.

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