Prove that for any odd integer a, $a^{33} \equiv a \pmod {4080}$

Question

Use Euler's theorem to prove that:

For any odd integer a, $a^{33} \equiv a \pmod {4080}.$

The hint given in the book is that $4080 = 15 \times 16 \times 17$, but I do not know how to use Euler's theorem to deal with $n = 15,17$,the condition $gcd (a,n) =1$ is not satisfied in those cases, could anyone clarify this for me please?

Answer 1

Euler's theorem for primes (Also known as Fermat's little theorem) tells you that $a^{p-1}\equiv 1\mod p$ if $\gcd(p,a)=1$.

For $p=3$ you have

$a^2\equiv 1\mod 3$ if $\gcd (a,3)=1$. So, $a^{33}\equiv (a^2)^{16}a\equiv a\bmod 3$ if$\gcd (a,3)=1$. But note that $a^{33}\equiv a\bmod 3$ also in the case when $3|a$ (in that case $a^{33}\equiv a\equiv 0\bmod 3$.)

So, $a^{33}\equiv a\mod 3$ for all $a$.

Do the same for $p=5$, $p=17$ and prove that $a^{33}\equiv a\mod 5$ and $a^{33}\equiv a\mod 17$.

Finally, since $a$ is odd, $\gcd(16,a)=1$. So, Euler's theorem tells you that $a^8\equiv 1\mod 16$. Conclude from here that $a^{33}\equiv a\mod 16$.

If you have $a^{33}\equiv a$ mod 3,5,16,17. Then you have your result.

Short answer: Since $\varphi(3)$,$\varphi(5)$,$\varphi(17)$,$\varphi(16)$ are divisors of $32$. Then $a^{33}\equiv a\mod(3\times 5\times 17\times 16).$

Answer 2

Using Fermat's Little Theorem

$17$ divides $a^{16m+1}-a$

$3$ divides $a^{2n+1}-a$

$5$ divides $a^{4p+1}-a$

So, we need $16m=2n=4p\implies n=8m,p=4m$

Using Carmichael Function as $a$ is odd $$a^{2^n}\equiv1\pmod{2^{n+2}}$$ for $n\ge2$

If $n=4\implies2^{4+2}$ divides $a^{16}-1$ which again divides $(a^{16})^r-1^r$

$\implies2^6$ divides $a^t-1$ if $16|t$

So, $a^{16m+1}-a$ will be divisible by lcm of $(3,5,17,2^6)$

Here $16m+1=33$