### Fermat's Little Theorem and Euler's Phi function

Fermat's Little Theorem states that if p is prime then a^p - a is a multiple of p. In other words,

a ^ (p-1) = 1 mod p, whenever a is not a multiple of p

The proof for this theorem is interesting. Take any number a that is co-prime with p. Now, a will correspond to some r (mod) p, where r lies between 1 and p-1

Now, take the numbers a, 2a, 3a, 4a, 5a,...(p-1)a...All these numbers will be co-prime with p (as p is prime) and all will leave remainders from 1 to p-1. Let us assume they leave remainders r, r

_{2},r_{3},r_{4, }…r_{p-1. }Now, none of these remainders will be equal to 0. Importantly, no two of these remainders can be equal.
(This is a critical result, which is established below)

Suppose r

_{3 }were equal to r_{5}_{. }Then, 5a -3a would be a multiple of p => 2a is a multiple of p, which is impossible.
This implies r, r

_{2},r_{3},r_{4, }…r_{p-1 }should correspond to 1,2,3,...p-1 in some order.
Now, when we multiply a, 2a, 3a, 4a,...(p-1)a in modular arithmetic terms, we should get a remainder of 1*2*3*...(p-1)

Or, a

^{p-1}* 1*2*3*...(p-1) = 1*2*3*...(p-1) mod p
Let us say 1*2*3*...(p-1) = X

Or, a

^{p-1}* X = X mod p
Or, X (a

^{p-1}-1) = 0 mod p
Now, X cannot be 0 mod p, so, a ^ (p-1) has to be 1 mod p, which is Fermat's Little Theorem

Now, for Euler's Phi function and Theorem.

Euler's Phi function states that

For two numbers m,n that are coprime (HCF of m,n =1)

m

^{phi(n)}= 1 mod n, (m^{phi(n) }leaves a remainder of 1 when divided by n)
Now, m and n are co-prime numbers. The possible remainders that m can have when divided by n are numbers from 0 to n-1 that are co-prime to n. A set that has phi(n) elements. We have seen this result here .

Now, let the elements in that set be R

_{1}R_{2},R_{3},R_{4, }…R_{p }set of possible remainders that m can leave when divided by n. This implies that p = phi(n).
Now, let us assume that m leaves a remainder R on division by n, R belongs to the above mentioned set.

Let us take the numbers R

_{1 }* m, R_{2}_{ }* m, R_{3}_{ }* m,R_{4}_{ }* m_{, }…R_{p}_{ }* m. Now, none of these would correspond to o mod n. Importantly, no two of these can be equal mod n either.
Because if

_{ }R_{4}_{ }* m and_{ }R_{7}_{ }* m were equal mod n, then m *( R_{4}_{ }- R_{7}) would be a multiple of n, which is impossible.
Therefore, the numbers R

_{1 }* m, R_{2}_{ }* m, R_{3}_{ }* m,R_{4}_{ }* m_{, }…R_{p}_{ }* m should correspond to the numbers R_{1 }, R_{2}_{ }, R_{3}_{ },R_{4}_{, }…R_{p }mod n in some order.
Now, if we take the product of the p numbers R

_{1 }* m, R_{2}_{ }* m, R_{3}_{ }* m,R_{4}_{ }* m_{, }…R_{p}_{ }* m. This will be equal to R_{1 }* R_{2}_{ }* R_{3}_{ }* R_{4}_{* }…* R_{p }mod n.
Let R

_{1 }* R_{2}_{ }* R_{3}_{ }* R_{4}_{* }…* R_{p }= Y
m

^{p}Y = Y mod n
Y (m

^{p}-1) = 0 mod n
Or, m

^{p}= 1 mod n
m ^ phi(n) = 1 mod n, which is Euler's Phi Function.

Brilliant Theorem. Beautiful implications. But, I dont think CAT will have a question based on this theorem.

In the same 'too tough for CAT category' one can file Wilson's theorem as well.

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Labels: CAT number theory, Euler's Phi Function, Fermat's Little Theorem

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