I am having trouble knowing where to start to use Fermat's Little Theorem to find the least residue of 5^10 (mod 11). Could you please offer some guidance? Thanks.
Hello, you want to solve this for $\displaystyle x$ :
$\displaystyle 5^{12} \equiv x \pmod{11}$
By Fermat's Little Theorem, we are left with $\displaystyle 5^2 \equiv x \pmod{11}$. Now, $\displaystyle 5^2 = 25$, and $\displaystyle 25 \equiv 3 \pmod{11}$, so $\displaystyle 5^{12} \equiv 3 \pmod{11}$.
Basically, you want to simplify the difficult calculation $\displaystyle 5^{12}$ until the remaining operations can be done mentally or easily on a calculator. It is up to you to decide whether $\displaystyle 5^2$ can be calculated manually.
Note that Fermat's Little Theorem is simply one particular case of Euler's Theorem :Remember though that Fermat's little Theorem only works if p is a prime number.
$\displaystyle a^{\varphi{(m)}} \equiv 1 \pmod{m}$ iff $\displaystyle gcd(a, m) = 1$. $\displaystyle \varphi{(m)}$ is the Euler totient function, that is, the quantity of numbers that are coprime with $\displaystyle m$. Coincidentally, if $\displaystyle m$ is prime, then $\displaystyle \varphi{(m)} = m - 1$