Question

What does Fermat’s little theorem says

Answer 1

Hint: To understand this theorem firstly get the overview on the prime numbers and how to check that the given number is divisible by a certain integer or not. After knowing this you can easily understand Fermat's little theorem and then taking some examples you can also verify the statement of the theorem.

Complete step-by-step solution:
Fermat’s little theorem gets the name from the mathematician who gives the statement of this theorem. This is the theorem given by the French mathematician Pierre de Fermat who was born in \[1607\]. He plays a very vital role in analytic geometry, probability and optics. He has contributed in physics as well as in mathematics. He is famous for Fermat's principle for light propagation in his Fermat's Last Theorem in number theory.
Now let us understand what Fermat says in his Fermat’s little theorem.
Fermat’s little theorem states that if \[p\] is a prime number, then for any integer \[a\], the number \[{{a}^{p}}-a\] is an integer multiple of \[p\]. If we express this in the mathematical for using modulus arithmetic, we can write it as
\[{{a}^{p}}\equiv a(\bmod p)\]
Here the modular operator or mod denotes that the given number wraps around after it will reach a certain value or designated value. This technique was developed by Carl Friedrich Gauss. Let us take the example to understand this concept more easily. There are \[24\] hours in a day, but in the wall clock or in watches there are only \[12\] hours. This means that after \[12\] the number wraps around.
Consider the example to get the clear picture of Fermat’s little theorem. Let us say \[a=2\] and \[p=7\], then by the statement of this theorem, we can say that \[{{a}^{p}}-a\] is equal to
\[\begin{align}
  & \Rightarrow {{2}^{7}}-2 \\
 & \Rightarrow 128-2 \\
 & \Rightarrow 126 \\
\end{align}\]
And we know that \[126\] is the multiple of \[7\].
There is a special case for the Fermat’s little theorem and that case is if the integer \[a\] is not divisible by a prime number \[p\] then the theorem statement is equivalent to the statement \[{{a}^{p-1}}-1\] is the integer which is the multiple of \[p\].

Note: Pierre de Fermat has given many contributions. His contribution leads to the infinitesimal calculus i.e. the branch of mathematics that deals with the continuous change in the given function. He is also known for discovering the method to find the smallest and greatest ordinates of the curved lines.