What is the units digit of 9^20? Step by Step

Step by step

The question

Question: what is the units digit of 9²0?

units digit of 9²⁰ = ?

1 Find the block

9 to the power 1 ends in 9.

9¹ = ends in 9

The ones digit of the next power comes only from the ones digit times 9.

So we keep just the ones digits.

only the ones digit matters: ones digit × 9

9 to the power 2: 9 times 9 is 81.

It ends in 1.

9² = 9 × 9 = 81: ends in 1

9 to the power 3: 1 times 9 is 9.

It ends in 9 again: we are back at the start.

9³ = 1 × 9 = 9: ends in 9

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

block = 9, 1

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 10 is 20.

So 10 full blocks take us to the 20th power.

20 minus 20 is 0 more.

20 ÷ 2 = 2 × 10 = 20, 20 − 20 = 0

3 Count into the block

Nothing is left over.

So the 20th is the end of a block: the last one, 1.

units digit of 9²⁰ = nothing left: the last in the block = 1

4 Check

Let's check by counting on.

A block ends at the 18th power.

Then we count on: 9, 1.

The 20th is 1.

So it is right!

18th = 1 (end of a block), 19th = 9, 20th = 1 ✓

Let's look back.

First: find the block.

Then: divide.

Then: count into the block.

Last: check.

Answer