Find the last digit of the number 2^555666777
Okay.
You are looking for the last digit of . You don't need all the other digits. So, you can consider your number modulo (which means you only consider the remainder of the number when divided by ). For instance, because divided by leaves a remainder of . And this gives us the last digit.
Now what I've shown you in my preceding post basically tells us that given :
- if then the last digit of is 6.
- if then the last digit of is 2.
- if then the last digit of is 4.
- if then the last digit of is 8.
And in your case, , and therefore the last digit of is .
Does it make sense ?
so i need to take 555666777/10 and since my remainder is 1, the last digit of my number ends in a 2. yes i do understand now thank you so much.
another question for clarification on my part. if i were taking 11^597637 and dividing it by 4, would the remainder of this always be 3, no matter what the exponent is?
No ! 555666777/4 gives a remainder of 1.so i need to take 555666777/10 and since my remainder is 1
The method I exposed here only works for some particular cases (here 2) because of special properties. But in the example you just gave above, using Euler's Theorem, given that , we can conclude that 11 raised to any power will give a remainder of 3 when divided by 4. But again, this is a little more advanced !if i were taking 11^597637 and dividing it by 4, would the remainder of this always be 3, no matter what the exponent is?
I encourage you to learn modulus (congruences) and all associated theorems asap, it really helps a lot in number theory.