Number Theory - Find Remainder when dividing by 17

[mahk_lolita]

Number Theory -- Find Remainder .. when dividing by 17

Homework Statement

Find the remainder when 3^24*5^13 is divided by 17.

Homework Equations

I know that 3^24 = 16 (mod 17)
and calculated that 5^13 mod 17 = 3 (mod 17)

The Attempt at a Solution

BUT, I'm completely unsure if I'm able to break up the products and take the modulo 17 of them separately.

What can I do? Help please!

 

[lanedance]

hey mahk lolita welcome to pf!

you should the information about the remainder of the componenents as follows
3^24 = a.17+16
5^13 = b.17+3
then
3^24*5^13 = (a.17+16)(b.17+3)

 

[mahk_lolita]

Thanks, lanedance!
With
(17a+16)(17b+3) = a sum whose parts have 17 as a factor... + 48 = 19 (mod 17.)

19 is the remainder.

So, I guess my question is, just to have a clear understanding, that you <i>can</i> split up the product? And by representing the number 3^24 and some sum (17a+16) and the same with 5^13, the answer will be the same? (Opposed to trying to calculate it directly with some powerful calculator.)

 

[lanedance]

yep!