Determine the orders of the following:
(a) $3 \pmod {10}$
(b) $7 \pmod {12}$
(c) $9 \pmod {16}$
(d) $11 \pmod {25}$
(e) $3 \pmod {13}$
(a) The following calculations show $3^4 \equiv 81 \equiv 1 \pmod {10}$.
| n | 3^n | 3^n mod 10 |
| 1 | 3 | 3 |
| 2 | 9 | 9 |
| 3 | 27 | 7 |
| 4 | 81 | 1 |
Since $\gcd(3,10)=1$, the order of 3 modulo 10 is 4.
(b) The following calculations show $7^2 \equiv 49 \equiv 1 \pmod {12}$.
| n | 7^n | 7^n mod 12 |
| 1 | 7 | 7 |
| 2 | 49 | 1 |
Since $\gcd(7,12)=1$, the order of 7 modulo 12 is 2.
(c) The following calculations show $9^2 \equiv 81 \equiv 1 \pmod {16}$.
| n | 9^n | 9^n mod 16 |
| 1 | 9 | 9 |
| 2 | 81 | 1 |
Since $\gcd(9,16)=1$, the order of 9 modulo 16 is 2.
(d) The following calculations show $11^5 \equiv 161051 \equiv 1 \pmod {25}$.
| n | 11^n | 11^n mod 25 |
| 1 | 11 | 11 |
| 2 | 121 | 21 |
| 3 | 1331 | 6 |
| 4 | 14641 | 16 |
| 5 | 161051 | 1 |
Since $\gcd(11,25)=1$, the order of 11 modulo 25 is 5.
(e) The following calculations show $3^3 \equiv 27 \equiv 1 \pmod {13}$.
| n | 3^n | 3^n mod 13 |
| 1 | 3 | 3 |
| 2 | 9 | 9 |
| 3 | 27 | 1 |
Since $\gcd(3,13)=1$, the order of 3 modulo 13 is 3.