What is the units digit of 13^22? Step by Step

Step by step

The question

Question: what is the units digit of 13²2?

units digit of 13²² = ?

1 Find the block

13 to the power 1 ends in 3.

13¹ = 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.

only the ones digit matters: ones digit × 3

13 to the power 2: 3 times 3 is 9.

It ends in 9.

13² = 3 × 3 = 9: ends in 9

13 to the power 3: 9 times 3 is 27.

It ends in 7.

13³ = 9 × 3 = 27: ends in 7

13 to the power 4: 7 times 3 is 21.

It ends in 1.

13⁴ = 7 × 3 = 21: ends in 1

13 to the power 5: 1 times 3 is 3.

It ends in 3 again: we are back at the start.

13⁵ = 1 × 3 = 3: ends in 3

So the ones digits repeat: 3, 9, 7, 1, again and again.

block = 3, 9, 7, 1

The block is 4 long.

length = 4

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

4 times 5 is 20.

So 5 full blocks take us to the 20th power.

22 minus 20 is 2 more.

22 ÷ 4 = 4 × 5 = 20, 22 − 20 = 2

3 Count into the block

Careful!

The 5 only counts full blocks.

The 2 left over gives the place.

Careful: not 5: the number of blocks ✗

2 left means place 2 in the block: 9.

units digit of 13²² = place 2 in the block = 9

4 Check

Let's check by counting on.

A block ends at the 20th power.

Then we count on: 3, 9.

The 22nd is 9.

So it is right!

20th = 1 (end of a block), 21st = 3, 22nd = 9 ✓

Let's look back.

First: find the block.

Then: divide.

Then: count into the block.

Last: check.

Answer