"Last digit in ∑k=1999km (olympiad question)" was answered by our community

Miguel Hanson

Answered question

2022-04-16

Last digit in

\sum_{k = 1}^{999} k^{m}

(olympiad question)

Gonarsu2dw8

Beginner2022-04-17Added 19 answers

Step 1
You can add

1000^{m}

to the sum as it will not change the last digit as its last digit is 0. Last digits of

1^{m}, 11^{m}, 21^{m}, \dots, 991^{m}

are the same. Similarly for

2^{m}, 12^{m}, \dots, 992^{m}

and so on till

10^{m}, 100^{m}, \dots {.1000}^{m}

. So the ones digit of

1^{m} + 2^{m} + \dots + 10^{m}

is the same as that of

11^{m} + 12^{m} + \dots + 20^{m}

and so on. There are 100 10s in 1000, so the ones digit of the sum is

(1^{m} + 2^{m} + \dots + 10^{m}) \times 100 mod 10

which is 0.

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?