Remainder when 2^100 is divided by 5

Step by step

The question

Question: find 2¹⁰⁰ mod 5.

2 to the power 100 has 31 digits.

Too big to work out by hand.

1 Small powers first

Let's make a table of small powers.

Each one is the one before times 2.

We only keep the remainder: the big part never matters.

2 to the 1 is 2.

Then 2 times 2 is 4.

4 mod 5 is 4.

And so on.

2 Find the pattern

At k = 5 we are back to 2.

From here everything repeats.

So the pattern 2, 4, 3, 1 repeats every 4 powers.

3 Where 100 lands

100 divided by 4: 25 full patterns, and 0 left over.

Careful!

Nothing left over means we end at the end of a pattern: the same place as k = 4.

So 2 to the 100 is like k = 4: the last one of the pattern.

So 2¹⁰⁰ mod 5 is 1.

4 Check

A computer can work out all 31 digits: 126765, and so on, 5376.

Divided by 5, it leaves 1, the same as our pattern.

Right!

Let's look back.

First: small powers first.

Then: find the pattern.

Then: where 100 lands.

Last: check.