Step by step
The question
Question: what is the ones digit of 3⁴7?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the block
3 to the power 1 ends in 3.
The ones digit of the next power comes only from the ones digit times 3.
So we keep just the ones digits.
3 to the power 2: 3 times 3 is 9.
It ends in 9.
3 to the power 3: 9 times 3 is 27.
It ends in 7.
3 to the power 4: 7 times 3 is 21.
It ends in 1.
3 to the power 5: 1 times 3 is 3.
It ends in 3 again: we are back at the start.
So the ones digits repeat: 3, 9, 7, 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 11 is 44.
So 11 full blocks take us to the 44th power.
47 minus 44 is 3 more.
3 Count into the block
Careful!
The 11 only counts full blocks.
The 3 left over gives the place.
3 left means place 3 in the block: 7.
4 Check
Let's check by counting on.
A block ends at the 44th power.
Then we count on: 3, 9, 7.
The 47th is 7.
So it is right!
5 Pick the choice
Our answer is 7.
That is choice C.
6 Why not the others
A is 1.
Wrong: that is the end of a block, but 47 ÷ 4 leaves 3.
B is 3.
Wrong: the ones digit of 3 itself: the power changes it.
D is 9.
Wrong: counted 2 into the block: 47 − 44 is 3.
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Then: check.
Then: pick the choice.
Last: why not the others.