Question

# Find the smallest positive integer k satisfying the stated condition or explain no such k exists: 3^k -= 1 mod 11.

Congruence
Find the smallest positive integer k satisfying the stated condition or explain no such k exists:
$$\displaystyle{3}^{{k}}\equiv{1}\text{mod}{11}$$.

2021-01-29
Step 1
Since gcd(3,11)=1 and again by using Euler's theorem,
$$\displaystyle{3}^{{\phi{\left({11}\right)}}}\equiv{1}{\left(\text{mod}{11}\right)}$$
$$\displaystyle{3}^{{10}}\equiv{1}{\left(\text{mod}{11}\right)}$$
So k can be 10 if no other smaller positive value than 10 satisfies the condition.
Step 2
The possible smaller positive values lesser than 10 satisfy the condition can be the divisors of 10.
Divisors of 10 less than 10 are 1, 2 and 5.
Check the congruences at these values.
$$\displaystyle{3}^{{1}}\equiv{3}{\left(\text{mod}{11}\right)}$$
$$\displaystyle{3}^{{2}}\equiv{9}{\left(\text{mod}{11}\right)}$$
$$\displaystyle{3}^{{4}}\equiv{4}{\left(\text{mod}{11}\right)}\Rightarrow{3}^{{5}}\equiv{1}{\left(\text{mod}{11}\right)}$$
So, 5 satisfies the condition of congruence.
Therefore, k=5.