Difficult Math

[Cherrybear]

Hi, I have an equation I need to solve. I need to figure out 3^2019÷5 because I need to know the remainder.
I already solved 3^2000÷5, and the remainder is 1, and 3^19, which has a remainder of 2.
All I need to know is how to put these together, and I know 3^2019=3^2000x3^19, but how exactly would I find the remainder from the two I have? If possible?
Thanks!

 

[Deleted member 4993]

Hi, I have an equation I need to solve. I need to figure out 3^2019÷5 because I need to know the remainder.
I already solved 3^2000÷5, and the remainder is 1, and 3^19, which has a remainder of 2.
All I need to know is how to put these together, and I know 3^2019=3^2000x3^19, but how exactly would I find the remainder from the two I have? If possible? Thanks!

I observe that the remainder comes in groups of (4, 2, 1, 3). What can you do with that information?

 

[pka]

Hi, I have an equation I need to solve. I need to figure out 3^2019÷5 because I need to know the remainder. I already solved 3^2000÷5, and the remainder is 1, and 3^19, which has a remainder of 2. All I need to know is how to put these together, and I know 3^2019=3^2000x3^19, but how exactly would I find the remainder from the two I have? If possible?

See here

 

[Dr.Peterson]

Hi, I have an equation I need to solve. I need to figure out 3^2019÷5 because I need to know the remainder.
I already solved 3^2000÷5, and the remainder is 1, and 3^19, which has a remainder of 2.
All I need to know is how to put these together, and I know 3^2019=3^2000x3^19, but how exactly would I find the remainder from the two I have? If possible?
Thanks!

You know these facts:

3^2000 has remainder 1, so that 3^2000 = 5m + 1​

3^19 has remainder 2, so that 3^19 = 5n + 2​

So 3^2019 = 3^2000 * 3^19 = (5m + 1)(5n + 2) = ...

What does that tell you about the remainder? (And what does it suggest about a general rule worth knowing?)

 

[Cherrybear]

Thanks everyone, you all helped a lot- I found that the remainder was 2, simply by multiplying the two numbers! (I don't know why I didn't think of that before XD)