Question

Just a quick question:

I'm trying to find the least residue of 7^5 mod 50.

I've been messing about with Chinese remainder theorem etc... But it seems a bit easy which is throwing me off and making me think I'm solving another question.

So instinctively I know that 7^2=49 so 49+1=50. Does that mean 7 is the least residue?? As in 7x7 = 49 +1 = 50?

Sorry this is very basic but this is my first go at modular arithmetic (well knowingly).

Thanks

Answer 1

I think you are having difficulty understanding the meaning of modular arithmetic
I will use an example
e.g. what is 15 mod 12
Think of a clock, after we have gone around counting the hours 1,2,3,... etc we start over again after 12 hours
so it becomes obvious that 15mod12 = 3
or
think of it this way:
What is the remainder when I divide 15 by 12 ?

so to our question:
7^5 mod 50
= 16807 mod 50 ---> 16807/50 = 336.14
so could take away 336(50) from 16807 to leave us with 16807 - 336(50) = 7

thus: 7^5 mod 50 = 7
=

Answer 2

Well op i feel like you might be doing the same TMA as me :'D

basically what I did is break down 7^5 into (7^2)*(7^3)

I then worked that out and -50

I got 1 and 13... obviously 1 is the least residual