Figuring the Last Digit of x^x^x^x

[bob j]

Hi All,
suppose I have something like x^x^x^x where x is an integer. Is there any trick to figure the last digit of the result?

Thank you

 

[disregardthat]

You can reduce x modulo 10, the exponent of x by modulo phi(10) = 4, the next exponent by phi(phi(10))=3, and the last by phi(phi(phi(10))=2, can you see why?

 

[bob j]

not really.. could you possibly give me a simple example?

 

[disregardthat]

I was a bit quick, but basically you make repeated use of Eulers theorem that says a^phi(n) = 1 mod n if gcd(a,n) = 1.

What you want is x^x^x^x mod 10. Say, 13^13^13^13. We can by elementary modulo arithmetic reduce the base mod 10, giving 3^13^13^13. By eulers theorem, we can reduce the exponent by phi(10) = 4 (do you understand this?). Thus we need to find 13^13^13 mod 4. Reducing base: 1^13^13=1, hence giving 13^13^13^13=3^1=3 mod 10.

But, if x is not relatively prime to 10, or phi(10), etc.. you must do some other tricks. Say you want to find 5^5^5^5 mod 10. You can however in this case use that 5^n = 5 mod 10 for all n, hence 5^5^5^5 = 5. This works for any multiple of 5. You can do the even numbers for yourself.

Just try to reduce as much as you can. Finding the last digit is almost always very easy.

 

[blue_raver22]

for your question! Yes, there is a trick to figuring out the last digit of x^x^x^x. It involves using modular arithmetic, which is a mathematical tool for finding the remainder when a number is divided by another number.

To find the last digit of x^x^x^x, we can use the fact that the last digit of a number only depends on the last digit of its powers. So, we can focus on finding the last digit of x^x, which we can then use to find the last digit of x^x^x and so on.

First, we can use modular arithmetic to find the remainder when x is divided by 10. This remainder will be our starting point for finding the last digit of x^x. For example, if x = 7, then the remainder when 7 is divided by 10 is 7.

Next, we can use the fact that the last digit of x^x will be the same as the last digit of x. In our example, 7^7 will have a last digit of 7.

We can then repeat this process for x^x^x and x^x^x^x, always using the last digit of the previous result. In our example, 7^7^7 will have a last digit of 3, and 7^7^7^7 will have a last digit of 7.

In summary, to find the last digit of x^x^x^x, we can use modular arithmetic to find the remainder when x is divided by 10, and then use the last digit of each subsequent power of x to find the final result. I hope this helps!