Step by step
The question
Question: find the last digit of 3⁹⁹.
3 to the power 99 has 48 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 3.
We only keep the remainder: the big part never matters.
3 to the 1 is 3.
Then 3 times 3 is 9.
9 mod 10 is 9.
And so on.
2 Find the pattern
At k = 5 we are back to 3.
From here everything repeats.
So the pattern 3, 9, 7, 1 repeats every 4 powers.
3 Where 99 lands
99 divided by 4: 24 full patterns, and 3 left over.
Careful!
The pattern is all we need.
The full number is far too big.
So 3 to the 99 is like k = 3.
So the last digit of 3⁹⁹ is 7.
4 Check
A computer can work out all 48 digits: 171792, and so on, 0667.
It ends in 7, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 99 lands.
Last: check.