What is the units digit of 4^31? Step by Step

Step by step

The question

Question: what is the units digit of 4³1?

units digit of 4³¹ = ?

1 Find the block

4 to the power 1 ends in 4.

4¹ = ends in 4

The ones digit of the next power comes only from the ones digit times 4.

So we keep just the ones digits.

only the ones digit matters: ones digit × 4

4 to the power 2: 4 times 4 is 16.

It ends in 6.

4² = 4 × 4 = 16: ends in 6

4 to the power 3: 6 times 4 is 24.

It ends in 4 again: we are back at the start.

4³ = 6 × 4 = 24: ends in 4

So the ones digits repeat: 4, 6, again and again.

block = 4, 6

The block is 2 long.

length = 2

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.

n ÷ (length of the block): what is left = the place in the block

2 times 15 is 30.

So 15 full blocks take us to the 30th power.

31 minus 30 is 1 more.

31 ÷ 2 = 2 × 15 = 30, 31 − 30 = 1

3 Count into the block

Careful!

The 15 only counts full blocks.

The 1 left over gives the place.

Careful: not 15: the number of blocks ✗

1 left means place 1 in the block: 4.

units digit of 4³¹ = place 1 in the block = 4

4 Check

Let's check by counting on.

A block ends at the 30th power.

Then we count on: 4.

The 31st is 4.

So it is right!

30th = 6 (end of a block), 31st = 4 ✓

Let's look back.

First: find the block.

Then: divide.

Then: count into the block.

Last: check.

Answer