What is the value of $4^{234}$ mod 11

Step 1 (Fermat's Little Theorem): Modulo $11$ (which is prime), Fermat's Little Theorem states that for any integer $a$ coprime to 11:

For $a=4$, we have $4^{10}\equiv 1(\mathrm{mod}11)$.

Step 2 (Exponent Division): We divide the exponent $234$ by $10$ to find the remainder:

Therefore:

Step 3 (Evaluate 4^4 mod 11):

• $4^2=16\equiv 5(\mathrm{mod}11)$

• $4^4=(4^2{)}^2\equiv 5^2=25(\mathrm{mod}11)$

• $25=2\times 11+3\equiv 3(\mathrm{mod}11)$

Step 4 (Conclusion): The value of $4^{234}(\mathrm{mod}11)$ is exactly 3.