Step by step
The question
Question: what is the ones digit of 2³0?
1 Find the block
2 to the power 1 ends in 2.
The ones digit of the next power comes only from the ones digit times 2.
So we keep just the ones digits.
2 to the power 2: 2 times 2 is 4.
It ends in 4.
2 to the power 3: 4 times 2 is 8.
It ends in 8.
2 to the power 4: 8 times 2 is 16.
It ends in 6.
2 to the power 5: 6 times 2 is 12.
It ends in 2 again: we are back at the start.
So the ones digits repeat: 2, 4, 8, 6, again and again.
The block is 4 long.
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
4 times 7 is 28.
So 7 full blocks take us to the 28th power.
30 minus 28 is 2 more.
3 Count into the block
Careful!
The 7 only counts full blocks.
The 2 left over gives the place.
2 left means place 2 in the block: 4.
4 Check
Let's check by counting on.
A block ends at the 28th power.
Then we count on: 2, 4.
The 30th is 4.
So it is right!
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.