What is the ones digit of 7^100? Step by Step

Step by step

The question

Question: what is the ones digit of 7¹00?

ones digit of 7¹⁰⁰ = ?

1 Find the block

7 to the power 1 ends in 7.

7¹ = ends in 7

The ones digit of the next power comes only from the ones digit times 7.

So we keep just the ones digits.

only the ones digit matters: ones digit × 7

7 to the power 2: 7 times 7 is 49.

It ends in 9.

7² = 7 × 7 = 49: ends in 9

7 to the power 3: 9 times 7 is 63.

It ends in 3.

7³ = 9 × 7 = 63: ends in 3

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

It ends in 1.

7⁴ = 3 × 7 = 21: ends in 1

7 to the power 5: 1 times 7 is 7.

It ends in 7 again: we are back at the start.

7⁵ = 1 × 7 = 7: ends in 7

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

block = 7, 9, 3, 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 25 is 100.

So 25 full blocks take us to the 100th power.

100 minus 100 is 0 more.

100 ÷ 4 = 4 × 25 = 100, 100 − 100 = 0

3 Count into the block

Nothing is left over.

So the 100th is the end of a block: the last one, 1.

ones digit of 7¹⁰⁰ = nothing left: the last in the block = 1

4 Check

Let's check by counting on.

A block ends at the 96th power.

Then we count on: 7, 9, 3, 1.

The 100th is 1.

So it is right!

96th = 1 (end of a block), 97th = 7, 98th = 9, 99th = 3, 100th = 1 ✓

Let's look back.

First: find the block.

Then: divide.

Then: count into the block.

Last: check.

Answer