Power Tower Puzzle Problem

[phantasmagoriun]

Consider an exponential tower of three thousand 7's.
What is the remainder when you divide the tower by 11?
Note that this notation means 7^(7^7) not (7^7)^7. So it's kinda like the Ackermann Function maybe?
The final answer must be given as a single integer in the range 0-10.


Anyone got any ideas?

 

[CRGreathouse]

Anyone got any ideas?

Yes, like the Ackerman function, but that doesn't matter.

7 mod 11 = ?
7^2 mod 11 = ?
7^3 mod 11 = ?
7^4 mod 11 = ?
7^5 mod 11 = ?
7^6 mod 11 = ?
...

 

[tacman]

This is a very interesting problem! Let's break it down to solve it.

First, we need to understand what an exponential tower is. An exponential tower is a way of representing repeated exponentiation. In this case, we have 7^(7^7), which means we are raising 7 to the power of 7, and then raising that result to the power of 7 again, and so on for a total of 3000 times.

Next, we need to understand what the remainder is when dividing by 11. The remainder is the number left over after dividing the tower by 11. For example, if we divide 10 by 3, we get a remainder of 1 because 3 goes into 10 three times with a remainder of 1.

Now, we can use the property of modular arithmetic to solve this problem. Modular arithmetic is a way of finding the remainder when dividing by a number. In this case, we will use the property that (a*b) mod n = ((a mod n) * (b mod n)) mod n.

Using this property, we can break down the tower into smaller parts. First, we can find the remainder of 7^7 when divided by 11. This can be done using a calculator or by hand, and the result is 10.

Next, we can find the remainder of 10^7 when divided by 11. Again, this can be done using a calculator or by hand, and the result is 1.

Now, we can apply the property of modular arithmetic to find the remainder of (7^7)^7 when divided by 11. This is equivalent to finding the remainder of 10^7 when divided by 11, which we already know is 1.

We can continue this process for the remaining 2998 exponentiations, always finding the remainder when dividing by 11. This will result in a final remainder of 1.

Therefore, the remainder when dividing the power tower of three thousand 7's by 11 is 1. We can express this as a single integer in the range of 0-10 as 1.

I hope this helps in solving the Power Tower Puzzle Problem!