Step by step
The question
Question: find the last digit of 7¹⁰⁰.
7 to the power 100 has 85 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 7.
We only keep the remainder: the big part never matters.
7 to the 1 is 7.
Then 7 times 7 is 49.
49 mod 10 is 9.
And so on.
2 Find the pattern
At k = 5 we are back to 7.
From here everything repeats.
So the pattern 7, 9, 3, 1 repeats every 4 powers.
3 Where 100 lands
100 divided by 4: 25 full patterns, and 0 left over.
Careful!
Nothing left over means we end at the end of a pattern: the same place as k = 4.
So 7 to the 100 is like k = 4: the last one of the pattern.
So the last digit of 7¹⁰⁰ is 1.
4 Check
A computer can work out all 85 digits: 323447, and so on, 0001.
It ends in 1, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 100 lands.
Last: check.