What is the ones digit of 8^17? Step by Step

Step by step

The question

Question: what is the ones digit of 8¹7?

ones digit of 8¹⁷ = ?

1 Find the block

8 to the power 1 ends in 8.

8¹ = ends in 8

The ones digit of the next power comes only from the ones digit times 8.

So we keep just the ones digits.

only the ones digit matters: ones digit × 8

8 to the power 2: 8 times 8 is 64.

It ends in 4.

8² = 8 × 8 = 64: ends in 4

8 to the power 3: 4 times 8 is 32.

It ends in 2.

8³ = 4 × 8 = 32: ends in 2

8 to the power 4: 2 times 8 is 16.

It ends in 6.

8⁴ = 2 × 8 = 16: ends in 6

8 to the power 5: 6 times 8 is 48.

It ends in 8 again: we are back at the start.

8⁵ = 6 × 8 = 48: ends in 8

So the ones digits repeat: 8, 4, 2, 6, again and again.

block = 8, 4, 2, 6

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 4 is 16.

So 4 full blocks take us to the 16th power.

17 minus 16 is 1 more.

17 ÷ 4 = 4 × 4 = 16, 17 − 16 = 1

3 Count into the block

Careful!

The 4 only counts full blocks.

The 1 left over gives the place.

Careful: not 4: the number of blocks ✗

1 left means place 1 in the block: 8.

ones digit of 8¹⁷ = place 1 in the block = 8

4 Check

Let's check by counting on.

A block ends at the 16th power.

Then we count on: 8.

The 17th is 8.

So it is right!

16th = 6 (end of a block), 17th = 8 ✓

Let's look back.

First: find the block.

Then: divide.

Then: count into the block.

Last: check.

Answer