Step by step
The question
Question: find the last digit of 9⁷⁷.
9 to the power 77 has 74 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 9.
We only keep the remainder: the big part never matters.
9 to the 1 is 9.
Then 9 times 9 is 81.
81 mod 10 is 1.
And so on.
2 Find the pattern
At k = 3 we are back to 9.
From here everything repeats.
So the pattern 9, 1 repeats every 2 powers.
3 Where 77 lands
77 divided by 2: 38 full patterns, and 1 left over.
Careful!
The pattern is all we need.
The full number is far too big.
So 9 to the 77 is like k = 1.
So the last digit of 9⁷⁷ is 9.
4 Check
A computer can work out all 74 digits: 299690, and so on, 8169.
It ends in 9, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 77 lands.
Last: check.