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Prove Fermat’s little theorem, which is given in Theorem 10.6. (Hint: Consider the sequence a1, a2, . . . What must happen, and how?)

THEOREM 10.6.

If p is prime and,aΖp+then.

Short Answer

Expert verified

Using the definition of Fermat’s little Theorem which states that ifp is a prime number, then for any integer a, the numberapa is an integer multiple of,p the above problem is solved.

Step by step solution

01

Understanding the theorem 

Statement: ifpbe a prime and then.ap-11(modp) HereΖp+ is defined as Ζp+={1,,p-1}and(p,a) is co-prime. It is also referred to as a miniature Fermat's theorem and is occasionally written as .ap-11(modp)

Take into account the firstp-1 positive multiple of.a

a,2a,3a,,(p-1)a


As the little Fermat’s theorem states(p,a)“is co-prime (that is, pis not exactly divisible bya)”. Suppose xaand ya are taken in such a way that, the modulo p of xa and ya are equal.

02

Continuing with the proof of the theorem 

Now, it can be said that .x=s(modp) So, p-1the multiples by a above are non-zero and distinct; that is, they must be congruent to a,2a,3a,,(p-1)ain the same order. Now, multiply all these congruence together and which gives:

a,2a,3a,,(p-1)a=1×2×3××(p-1)(modp)

an-1(p-1)!=(p-1)!(modp)

Now, dividing each side by (p-1)!in the above equality.ap-11(modp)

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