Modular Arithmetic Proof with exponents

[JPanthon]

Homework Statement

Let p be a prime number.
Prove:

(a+b)^p modp = [(a^p modp) + (b^p modp)]modp

Homework Equations

modular arithmetic.

The Attempt at a Solution

I honestly haven't the slightest clue.
Would induction be my best bet here?
If so, when I suppose the statement is true for (k+1), n isn't always prime anymore.

I used to be a biochem major and just switched into algebra, so I'm sorry if I seem retarded, I'm just very behind! Help please!

 

[Dick]

Homework Statement

Let p be a prime number.
Prove:

(a+b)^p modp = [(a^p modp) + (b^p modp)]modp

Homework Equations

modular arithmetic.

The Attempt at a Solution

I honestly haven't the slightest clue.
Would induction be my best bet here?
If so, when I suppose the statement is true for (k+1), n isn't always prime anymore.

I used to be a biochem major and just switched into algebra, so I'm sorry if I seem retarded, I'm just very behind! Help please!

No, induction isn't your best bet for the very good reason you mention. Think about the binomial expansion of (a+b)^p.

 

[Dick]

Write the binomial expansion without knowing what p is. Just write it symbolically. Can you show many of the binomial coefficients are divisible by p?