Question

What is the ones digit in the number 2^2058. Hint: Start with the smaller exponents to find pattern

Answer 1

`4`

1) We need to find out a pattern for the powers that have a base 2:

`\begin{gathered} 2^(1)=2 \\ 2^2=4 \\ 2^3=8 \\ 2^4=16 \\ 2^5=32 \\ 2^6=64 \\ 2^7=128 \\ 2^8=256 \\ (\ldots) \end{gathered}`

Note that from 4 to 4 powers the last digit starts to repeat itself.

2) So, let's proceed with this dividing the exponent 2058 by 4:

Now, note that the remainder is 2, therefore we can state that:

`2^(2058)=same\: ones\: digit\: as\colon2^2=4`