Calculate 2^50 mod 7

Step by step

The question

Question: find 2⁵⁰ mod 7.

2 to the power 50 has 16 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 7 is 4.

And so on.

2 Find the pattern

At k = 4 we are back to 2.

From here everything repeats.

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

3 Where 50 lands

50 divided by 3: 16 full patterns, and 2 left over.

Careful!

The pattern is all we need.

The full number is far too big.

So 2 to the 50 is like k = 2.

So 2⁵⁰ mod 7 is 4.

4 Check

A computer can work out all 16 digits: 112589, and so on, 2624.

Divided by 7, it leaves 4, the same as our pattern.

Right!

Let's look back.

First: small powers first.

Then: find the pattern.

Then: where 50 lands.

Last: check.