Step by step
The question
Question: find the last digit of 8¹²³.
8 to the power 123 has 112 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 8.
We only keep the remainder: the big part never matters.
8 to the 1 is 8.
Then 8 times 8 is 64.
64 mod 10 is 4.
And so on.
2 Find the pattern
At k = 5 we are back to 8.
From here everything repeats.
So the pattern 8, 4, 2, 6 repeats every 4 powers.
3 Where 123 lands
123 divided by 4: 30 full patterns, and 3 left over.
Careful!
The pattern is all we need.
The full number is far too big.
So 8 to the 123 is like k = 3.
So the last digit of 8¹²³ is 2.
4 Check
A computer can work out all 112 digits: 120245, and so on, 7712.
It ends in 2, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 123 lands.
Last: check.