Question

Find the last two didgits of 6 raised to the power of 13
A) 96
B) 36
C) 76
D)16

Answer 1

6^13 = 13060694016
now guess your answer

Answer 2

? ?

Answer 3

is this number theory?

Answer 4

In number theory they ask this question but we cant use technology.

Answer 5

13060694016
which are the last two digits of the above number

Answer 6

yes, but you used a calculator...

Answer 7

there is a way to do it without.

Answer 8

Sorry , I get it now :)

Answer 9

Thanks guys :3

Answer 10

welcome

Answer 11

ok looks like this is fine:) good because I forgot how to do it with mod10

Answer 12

There is a pattern to observe:$$6^1=\color{red}6\\6^2=\color{blue}3\color{red}6\\6^3=2\color{blue}1\color{red}6\\6^4=12\color{blue}9\color{red}6\\6^5=67\color{blue}7\color{red}6\\6^6=406\color{blue}5\color{red}6\\6^7=2439\color{blue}3\color{red}6\\6^8=15036\color{blue}1\color{red}6$$
The last digit is *always* 6 and the 10's digit follows a regular pattern: \(3\to1\to9\to7\to5\to3\to1\to9\to7\to5\to3\to1\to\dots\)

Notice this pattern repeats a number every 5 digits; this means exponents that are a multiple \(5\) apart will have the same second digit. What about \(13\)? Well:$$6^{13}=6^{8+5}=6^{3+5+5}$$... so \(6^{13}\) ought to have the same last digits as \(6^8\) and \(6^3\), which we can see is \(16\).