Step by step
The question
Question: find the last digit of 4⁷⁷⁷.
4 to the power 777 has 468 digits.
Too big to work out by hand.
The last digit is the remainder after dividing by 10.
So we work mod 10.
1 Small powers first
Let's make a table of small powers.
Each one is the one before times 4.
We only keep the remainder: the big part never matters.
4 to the 1 is 4.
Then 4 times 4 is 16.
16 mod 10 is 6.
And so on.
2 Find the pattern
At k = 3 we are back to 4.
From here everything repeats.
So the pattern 4, 6 repeats every 2 powers.
3 Where 777 lands
777 divided by 2: 388 full patterns, and 1 left over.
Careful!
The pattern is all we need.
The full number is far too big.
So 4 to the 777 is like k = 1.
So the last digit of 4⁷⁷⁷ is 4.
4 Check
A computer can work out all 468 digits: 631848, and so on, 1984.
It ends in 4, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 777 lands.
Last: check.