Step by step
The question
Question: find 3⁵⁰ mod 11.
3 to the power 50 has 24 digits.
Too big to work out by hand.
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 11 is 9.
And so on.
2 Find the pattern
At k = 6 we are back to 3.
From here everything repeats.
So the pattern 3, 9, 5, 4, 1 repeats every 5 powers.
3 Where 50 lands
50 divided by 5: 10 full patterns, and 0 left over.
Careful!
Nothing left over means we end at the end of a pattern: the same place as k = 5.
So 3 to the 50 is like k = 5: the last one of the pattern.
So 3⁵⁰ mod 11 is 1.
4 Check
A computer can work out all 24 digits: 717897, and so on, 0249.
Divided by 11, it leaves 1, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 50 lands.
Last: check.