What is the Remainder of Dividing 2(562009)-3?

[bphysics]

Homework Statement

Find the remainder of dividing 2(56²⁰⁰⁹)-3.

Homework Equations

Let m be a positive integer. If a$$\equiv$$b (mod m) and c$$\equiv$$d (mod m), then a + c $$\equiv$$ b + d (mod m) and ac$$\equiv$$bd (mod m).

The Attempt at a Solution

Using ac$$\equiv$$bd (mod m):

(2 mod 55)(56²⁰⁰⁹mod 55) - (3 mod 55)

Using a + c $$\equiv$$ b + d (mod m)

(2 mod 55)((55²⁰⁰⁹ mod 55) + (1²⁰⁰⁹ mod 55)) - (3 mod 55)

(2)(0+1)-(3) = -1 OR remainder of 54

This was a problem on my math test and I got 52 as the remainder at first, but it was wrong.

Thx if you can help me.

 

[Dick]

54 is right, 52 is wrong. But your method is dubious. It looks like you are trying use a rule like (a+b)^n mod m=(a^n mod m)+(b^n mod m). That's not right. What is true is that a^n mod m=(a mod m)^n. Just use that 56 mod 55=1.

 

[bphysics]

Thanks so much =D, i'll remember that.