So, starting with my first daily note, the topic is Fermat's Little Theorem.
Topic: Number Theory
Fermat's Little Theorem states that
"For all natural numbers 'a' , , where 'p' is a prime number. "
Let us prove this out.
Consider the binomial expansion for the prime 'p',
But since, . So, . This implies that, .
Generalizing this we get, . By taking , we get . That's it, we got the result.
Phewwwwwww!! We have proved it.
Fermat's Theorem is very useful in some problems based on Modular Arithmetic.
Now, if ,i.e. if and are coprime to each other, then . This is known as Fermat's Little Theorem and it is a special case of Euler's Totient Theorem.
Now let us solve some problems.
Problem 1(introductory): Find the remainder when is divided by .
Solution: Observe that and from Fermat's Theorem . But . So, .
So, . This is can be even very easily using Fermat's littile theorem(Try Yourselves).
Problem 2: Find the remainder when is divided by .
Solution: Observe that So, , , , and . So, .
So, I think that I have given a clear picture on Fermat's Theorem. So, stay tuned for upcoming DAILY NOTES.