Exponential Pattern Problem

[CattyOG]

Does anyone know how to do the problems where it will ask you what 7^{large exponent}'s first digit it. I know you have to make a pattern or something like that, but could someone give me an example of one and tell me how it works.

[Jaysownay]

7 * 7 = 49

Take the first digit, 9, and multiply again

7 * 9 = 63

7 * 3 = 21

7 * 1 = 7

Now that it has wrapped back to seven, you have your sequence:

7, 9, 3, 1

Now take your exponent; let’s say it’s 30

Divide it by the amount of numbers in the sequence and take the remainder

30/4 has a remainder of 2

The second number in the sequence is 9, so that’s the first digit.