Step by step
The question
Question: what is the ones digit of 7⁴2?
1 Find the block
7 to the power 1 ends in 7.
The ones digit of the next power comes only from the ones digit times 7.
So we keep just the ones digits.
7 to the power 2: 7 times 7 is 49.
It ends in 9.
7 to the power 3: 9 times 7 is 63.
It ends in 3.
7 to the power 4: 3 times 7 is 21.
It ends in 1.
7 to the power 5: 1 times 7 is 7.
It ends in 7 again: we are back at the start.
So the ones digits repeat: 7, 9, 3, 1, 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 10 is 40.
So 10 full blocks take us to the 40th power.
42 minus 40 is 2 more.
3 Count into the block
Careful!
The 10 only counts full blocks.
The 2 left over gives the place.
2 left means place 2 in the block: 9.
4 Check
Let's check by counting on.
A block ends at the 40th power.
Then we count on: 7, 9.
The 42nd is 9.
So it is right!
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.