Determine the order of the following:
(a) $2 \pmod {11}$
(b) $3 \pmod {11}$
(c) $5 \pmod {11}$
(d) $7 \pmod {11}$
(a) $\gcd(11,2)=1$ and so the order exists.
The order is a factor of $\phi{11}=10$. These factors are 1, 2, 5, 10.
- $2^2 \equiv 4 \not \equiv 1 \pmod {11}$
- $2^5 \equiv 10 \not \equiv 1 \pmod {11}$
- $2^{10} \equiv 1 \pmod {11}$
The order of 2 modulo 11 is 10.
(b) $\gcd(3,11)=1$ and so the order exists.
The order is a factor of $\phi{11}=10$. These factors are 1, 2, 5, 10.
- $3^2 \equiv 9 \not \equiv 1 \pmod {11}$
- $3^5 \equiv 1 \pmod {11}$
The order of 3 modulo 11 is 5.
(c) $\gcd(5,11)=1$ and so the order exists.
The order is a factor of $\phi{11}=10$. These factors are 1, 2, 5, 10.
- $5^2 \equiv 3 \not \equiv 1 \pmod {11}$
- $5^5 \equiv 1\equiv 1 \pmod {11}$
The order of 5 modulo 11 is 5.
(d) $\gcd(7,11)=1$ and so the order exists.
The order is a factor of $\phi{11}=10$. These factors are 1, 2, 5, 10.
- $7^2 \equiv 5 \not \equiv 1 \pmod {11}$
- $7^5 \equiv 10 \not \equiv 1 \pmod {11}$
- $7^{10} \equiv 1 \pmod {11}$
The order of 7 modulo 11 is 10.