Step by step
The question
Question: find the last digit of 2⁵⁰.
2 to the power 50 has 16 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 2.
We only keep the remainder: the big part never matters.
2 to the 1 is 2.
Then 2 times 2 is 4.
4 mod 10 is 4.
And so on.
2 Find the pattern
At k = 5 we are back to 2.
From here everything repeats.
So the pattern 2, 4, 8, 6 repeats every 4 powers.
3 Where 50 lands
50 divided by 4: 12 full patterns, and 2 left over.
Careful!
The pattern is all we need.
The full number is far too big.
So 2 to the 50 is like k = 2.
So the last digit of 2⁵⁰ is 4.
4 Check
A computer can work out all 16 digits: 112589, and so on, 2624.
It ends in 4, the same as our pattern.
Right!
Let's look back.
First: small powers first.
Then: find the pattern.
Then: where 50 lands.
Last: check.