The one's digit in 2^4,000?

Question

What is the ones or units digit in 2^4,000? Could you please explain?

Answer 1

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16 (one's digit 6)

2^5 = 32 (one's digit 2)

So we see that the one's digit goes 2,4,8,6 and then repeats every fourth power.  Since 4000 is divisible by 4, it'll have the same one's place as 2^4, which we showed to be 6 above.

Answer: 6.

Answer 2

just look at the pattern:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128

2^8 = 256

2^9 = 512

See the pattern?

1 - 5 ..... ends in 2 ............ 3937

2 - 6 ..... ends in 4 ............ 3938

3 - 7 ..... ends in 8 ............ 3999

4 - 8 ......ends in 6 ............ 4000

They each just go up by 4 and 4 factors directly into 4000 sothe ones digit would be 6

Answer 3

Let's look at the first few powers of 2:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128

2^8 = 256

Notice that the units digits follow the pattern 2, 4, 8, 6, 2, 4, 8, 6, ...

Therefore, when 2 is raised to a multiple of 4, the resulting units digit is 6, so we can conclude that the units digit of 2^(4000) is 6.