What is the ones digit of 3^25? Step by Step

Step by step

The question

Question: what is the ones digit of 3²5?

ones digit of 3²⁵ = ?

1 Find the block

3 to the power 1 ends in 3.

3¹ = 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

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

It ends in 9.

3² = 3 × 3 = 9: ends in 9

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

It ends in 7.

3³ = 9 × 3 = 27: ends in 7

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

It ends in 1.

3⁴ = 7 × 3 = 21: ends in 1

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

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

3⁵ = 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 6 is 24.

So 6 full blocks take us to the 24th power.

25 minus 24 is 1 more.

25 ÷ 4 = 4 × 6 = 24, 25 − 24 = 1

3 Count into the block

Careful!

The 6 only counts full blocks.

The 1 left over gives the place.

Careful: not 6: the number of blocks ✗

1 left means place 1 in the block: 3.

ones digit of 3²⁵ = place 1 in the block = 3

4 Check

Let's check by counting on.

A block ends at the 24th power.

Then we count on: 3.

The 25th is 3.

So it is right!

24th = 1 (end of a block), 25th = 3 ✓

Let's look back.

First: find the block.

Then: divide.

Then: count into the block.

Last: check.

Answer