Do you need to know the digits of $\pi$ ?

$\Huge{3^{4k}\equiv 1 \pmod{10}}$

Same

Consider $3^{14^n} \bmod 10$ , where $n \ge 2$

Solution 1: By binomial expansion

$\begin{aligned} 3^{14^n} & \equiv (3^2)^{2^{n-1}\times 7^n} \text{ (mod 10)} \\ & \equiv 9^{2^{n-1}\times 7^n} \text{ (mod 10)} \\ & \equiv (10-1)^{2^{n-1}\times 7^n} \text{ (mod 10)} \\ & \equiv (-1)^\blue{2^{n-1}\times 7^n} \text{ (mod 10)} & \small \blue{\text{The exponent }2^{n-1}\times 7^n \text{ is even}} \\ & \equiv \boxed 1 \text{ (mod 10)} \end{aligned}$

Solution 2: By Euler's theorem . Since $\gcd(3,10) = 1$ , we can apply Euler's theorem as follows. Note that Euler's totient function $\phi(10) = 10 \times \frac 12 \times \frac 45 = 4$ .

$\begin{aligned} 3^{14^n} & \equiv 3^{14^n \bmod \phi(10)} \text{ (mod 10)} \\ & \equiv 3^{14^n \bmod 4} \text{ (mod 10)} \\ & \equiv 3^0 \text{ (mod 10)} \\ & \equiv \boxed 1 \text{ (mod 10)} \end{aligned}$

Last digit of 3^4 = 1.

1^any positive integer is always 1