Modular Arithmetic Proof with exponents

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SUMMARY

The discussion centers on proving the equation (a+b)^p mod p = [(a^p mod p) + (b^p mod p)] mod p, where p is a prime number. Participants suggest using the binomial expansion to demonstrate that many binomial coefficients are divisible by p, which is crucial for the proof. Induction is deemed inappropriate due to the nature of prime numbers. The conversation highlights the importance of understanding modular arithmetic and binomial coefficients in this context.

PREREQUISITES
  • Understanding of modular arithmetic
  • Familiarity with binomial expansion
  • Knowledge of prime numbers and their properties
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of binomial coefficients in modular arithmetic
  • Learn about Fermat's Little Theorem and its applications
  • Explore proofs involving modular exponentiation
  • Practice problems on binomial expansion and modular proofs
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Students of algebra, mathematicians interested in number theory, and anyone looking to deepen their understanding of modular arithmetic and its applications in proofs.

JPanthon
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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!
 
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JPanthon said:

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.
 
Last edited:
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?
 

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