Step by step
The question
Question: find 5⁴⁰ mod 6.
5 to the power 40 has 28 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 5.
We only keep the remainder: the big part never matters.
5 to the 1 is 5.
Then 5 times 5 is 25.
25 mod 6 is 1.
And so on.
2 Find the pattern
At k = 3 we are back to 5.
From here everything repeats.
So the pattern 5, 1 repeats every 2 powers.
3 Where 40 lands
40 divided by 2: 20 full patterns, and 0 left over.
Careful!
Nothing left over means we end at the end of a pattern: the same place as k = 2.
So 5 to the 40 is like k = 2: the last one of the pattern.
So 5⁴⁰ mod 6 is 1.
4 Check
A computer can work out all 28 digits: 909494, and so on, 0625.
Divided by 6, it leaves 1, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 40 lands.
Last: check.