Find the units digit of $3^{777}$.

$$\small{\text{ Find the units digit of $3^{777}$ or $3^{777} = x \pmod {10} $ }}$$

$$\small{\text{$ \begin{array}{lrcll} 1. & ~~gcd(10,3) &=& 1 \qquad & | \qquad gcd = \mathrm{greatest~common~ divisor}\\ 2. & ~~\varphi{(10)} &=& 4 \qquad & | \qquad \varphi = \mathrm{Euler ~function}\qquad \varphi{(10)} = 10 \cdot (1-\dfrac{1}{2})\cdot (1-\dfrac{1}{5}) \\ 3. & ~~a^{\varphi (n)} &\equiv& 1 \pmod{n} \qquad & | \qquad \mathrm{Euler's ~Theorem,~~ if~ } gcd(a,n)=1 \qquad \\ & ~~ 3^4 &\equiv& 1 \pmod {10} \qquad & | \qquad n = 10\qquad a = 3\\\\ & & 3^{777} \pmod{10}\\\\ & &\equiv& 3^{4\cdot 194 +1} \pmod{10}\\ & &\equiv& 3^{4\cdot 194}\cdot 3 \pmod{10}\\ & &\equiv& (3^4)^{194}\cdot 3 \pmod{10}\\ & &\equiv& (\underbrace{3^4}_{=1 \pmod{10}})^{194}\cdot 3 \pmod{10}\\\\ & &\equiv& 1^{194}\cdot 3 \pmod{10}\\ & &\equiv & \textcolor[rgb]{1,0,0}{3} \pmod{10}\\ \\ & \mathbf{the~ Units~ digit~ of ~} 3^{777} \mathbf{ ~is ~} 3\\ \end{array} $ }}$$

When looking for the units digit of 3ⁿ, look at the value of mod(n,4).  When this is 0, the units digit is 1, when it is 1 the units digit is 3, when it is 2 the units digit is 9, when it is 3 the units digit is 7.

mod(777,4) = 1, so the units digit of 3⁷⁷⁷ is 3.

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