Daily Note #1 - Fermat's Little Theorem

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' , \(a^{p} \equiv a \mod p\) , where 'p' is a prime number. "

Let us prove this out.

Consider the binomial expansion for the prime 'p',

$(a+b)^{p} = a^{p} + \binom{p}{1} a^{p-1} b +\binom{p}{2} a^{p-2} b^{2} + ....... + b^{p}$.

But since, $p| \binom{p}{k} \forall k=1,2,3,.....p-1$. So, $\binom{p}{1} a^{p-1} b +\binom{p}{2} a^{p-2} b^{2} + ....... + \binom{p}{p-1} a b^{p-1} = Multiple(p)$. This implies that, $(a+b)^{p} \equiv a^{p} + b^{p} \mod p$.

Generalizing this we get, $(a_{1}+a_{2} +.....a_{n})^{p} \equiv a_{1}^{p} + a_{2}^{p} +........ +a_{n}^{p} \mod p$. By taking $a_{1}=a_{2}=.....=a_{n}=1$, we get $n^{p} \equiv n \mod p$. 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 $(a,p) =1$,i.e. if $a$ and $p$ are coprime to each other, then $a^{p-1} \equiv 1 \mod p$. 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 $37^{1123}$ is divided by $17$.

Solution: Observe that $37 \equiv 3 \mod 17$ and from Fermat's Theorem $37^{17} \equiv 37 \mod 17$. But $1123 = 66 \times 17 + 1$. So, $37^{1123} \equiv (37^{17})^{66} \times 37 \equiv 37^{66} \times 37 \equiv 37^{67} \equiv (37^{17})^{3} \times 37^{16}$ $\equiv 37^{3} \times 37^{16} \equiv 37^{17} \times 37^{2} \equiv 37 \times 37^{2} \equiv 37^{3} \equiv 3^{3} \equiv 27 \equiv 10 \mod 17$.

So, $37^{1123} \equiv 10 \mod 17$. This is can be even very easily using Fermat's littile theorem(Try Yourselves).

Problem 2: Find the remainder when $2^{20} + 3^{30} + 4^{40} + 5^{50} + 6^{60}$ is divided by $7$.

Solution: Observe that $(2,7)=(3,7)=(4,7)=(5,7)=(6,7)=1.$ So, $2^{20} \equiv 2^{2} \mod 7$, $3^{30} \equiv 1 \mod 7$, $4^{40} \equiv 4^{4} \equiv 4 \mod 7$, $5^{50} \equiv 5^{2} \equiv 4 \mod 7$ and $6^{60} \equiv 1 \mod 7$. So, $2^{20} + 3^{30} + 4^{40} + 5^{50} + 6^{60} \equiv 4+1+4+4+1 \equiv 0 \mod 7$.

So, $2^{20} + 3^{30} + 4^{40} + 5^{50} + 6^{60} \equiv 0 \mod 7$.

So, I think that I have given a clear picture on Fermat's Theorem. So, stay tuned for upcoming DAILY NOTES.