What is the reminder of 999^{1000 }divided by 5?

**Answer**

We first have to determine the possible units digits of powers of 9.

9^{1} = 9

9^{2} = 81

9^{3} = 729

9^{4} = 6561

9^{5} = 59049

We notice that the units digits of powers of 9 alternate between 1 and 9. For even powers of 9, the units digit is 1. For odd powers of 9, the units digit is 9.

Therefore the last digit of 999^{1000 }is 1.

Why do we only care about the last digit? Natural numbers that end in the same last digit have the same remainder when divided by 5.

A number that ends in 1, no matter what its other digits are, will have a remainder of 1 when divided by 5.

The reminder of 999^{1000 }divided by 5 is just 1.

Back to *Math of the Day*

### Like this:

Like Loading...