Towers of integers and divisibilty by 11

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In summary, The conversation discusses a problem involving determining whether a large number is divisible by 11. The person is trying to use modulo 11 to solve the problem, but is struggling with understanding how to apply it to the "tower" of exponents. They receive a hint that the number is not divisible by 11, and are given a pattern for working out powers of 10 and 5 (mod 11). By applying this pattern and simplifying the terms, it is determined that the number is indeed divisible by 11. The person shares their explanation of the problem and asks for feedback on their wording.
  • #1
scottyk
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I'm really stuck on the following problem:

I'm trying to determine whether or not

5^10^5^10^5 is divisible by 11... i have tried a few different methods and can't figure this out.

I know the trick must have something to do with modulo 11, but I am not sure exactly how to get the result.

Any help would be GREATLY appreciated. Thanks in advance.
 
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  • #2
I can tell you just from looking at it that it's not divisible by 11. Hint: the prime factorization has to include 11 to be divisible by 11, and your number is 5^...
 
  • #3
but is there a way to break down the "tower" into a smaller integer using modulo 11?

i know that indeed this integer is not divisible by 11, but many students are confused by the tower.

For example, 21 is equivalent to -2 for mod 23.

Is there a way to find that 5^10^5^10^5 is equivalent to some integer for mod 11?
 
  • #4
5 is raised to a power that is an integer. Therefore, the factorization of your number contains only factors of 5 (a LOT of factors of 5 but only factors of 5). That is equivalent to 0 mod 11.

I don't see the source of confusion.
 
  • #5
Ok, so let's take it one step further:

would
10^5^10^5^10 + 5^10^5^10^5 be divisble by 11?

i know that each tower is not divisble by 11, but what about when you add them together?

i'm not sure why this is confusing me so much...maybe I'm making it too complicated?
 
  • #6
What is the order of 5, or ten, mod 11?

Failing that, just work out 5^10 mod 11, then raise that to the power 5, mod it by 11, then rinse and repeat.10 is easy since it is -1 mod 11, so in your example 10^junk =1 mod 11

of course anything to the power p-1 mod p a prime is easy to work out by Fermat't little theorem.
 
  • #7
Here is how i plan to write up the problem:

We notice that powers of 10 (mod 11) form a pattern:

10^0 mod 11 = 1
10^1 mod 11 = 10
10^2 mod 11 = 1
10^3 mod 11 = 10
.
.
.
So all the odd powers of 10 are equal to 10 (mod 11).

Clearly, the exponent of the first term is odd: it's some power of 5.



Now, we also notice that powers of 5 (mod 11) form a pattern:

5^0 mod 11 = 1
5^1 mod 11 = 5
5^2 mod 11 = 3
5^3 mod 11 = 4
5^4 mod 11 = 9
.....
5^5 mod 11 = 1, and the above pattern continues.

So, if the exponent is a multiple of 5, we'll get a 1. Well, the
exponent is a multiple of 10, so we do get a one.

Therefore, 10^5^10^5^10 mod 11 = 10 + 5^10^5^10^5 mod 11 = 1

So, 10 + 1 mod 11 = 0.

Then we see that this integer is divisible by 11.




Ok, that's what I have. It seems right, but some of the
wording/explanations seem a little awkward to me. Any comments on
this would be greatly appreciated!
 
  • #8
You please see whether you have written what you mean in the post #1.
Do you mean to add the powers or power the powers?
 
  • #9
add the two "towers" of exponents


thats is 5^... + 10^...
 
  • #10
addingtwice = multiplying with two.
Your terms is 5^10^5^10, and not something like 5^10 + 10^5 and all such complicated things. The terms can be simply expressed as the product of two primes 5 and 2 with some powers.
 

What are towers of integers?

Towers of integers refer to a mathematical concept where a series of numbers are stacked on top of each other, starting with a single digit at the top and increasing in length as you go down. For example, a tower of integers for the number 123 would look like this:
1
12
123.

How do you determine if a tower of integers is divisible by 11?

To determine if a tower of integers is divisible by 11, you can use the "divisibility by 11" rule, which states that if the difference between the sum of the alternate digits (starting from the right) is either 0 or a multiple of 11, then the number is divisible by 11. For example, in the tower of integers for 123, the sum of the alternate digits would be 2, which is a multiple of 11, therefore the number is divisible by 11.

What is the significance of divisibility by 11 in towers of integers?

Divisibility by 11 in towers of integers is significant because it is a way to determine if the whole number represented by the tower is also divisible by 11. This can be useful in various mathematical and scientific calculations.

Are there any exceptions to the "divisibility by 11" rule in towers of integers?

Yes, there are a few exceptions to the "divisibility by 11" rule. These include numbers that have a difference between the sum of the alternate digits that is negative or a multiple of 11. In these cases, the number is not divisible by 11.

How can towers of integers and divisibility by 11 be applied in real-world situations?

Towers of integers and divisibility by 11 can be applied in many real-world situations, such as in banking and finance, where numbers must be checked for accuracy and divisibility by 11 can help identify errors. It can also be used in coding and computer science, as well as in various scientific fields that involve large numbers and calculations.

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