Part (a): Finding the Order of 2 and 3 Modulo 23

To find the order of an integer $a$ modulo $n$, we look for the smallest positive integer $k$ such that:

$a^k \equiv 1 \pmod{n}$

Step 1: Find the order of 2 modulo 23.

We need to find the smallest $k$ such that:

$2^k \equiv 1 \pmod{23}$

Since 23 is prime, Euler's Theorem tells us that the order of any integer modulo 23 must divide $\phi(23) = 22$. Therefore, $k$ must be a divisor of 22. The divisors of 22 are: $1, 2, 11, 22$.

We check the powers of 2 modulo 23:

• $2^1 = 2$
• $2^2 = 4$
• $2^{11} = 2048 \equiv 1 \pmod{23}$

Thus, the order of 2 modulo 23 is 11.

Step 2: Find the order of 3 modulo 23.

Now, we need to find the smallest $k$ such that:

$3^k \equiv 1 \pmod{23}$

Again, $k$ must divide $22$. We check the powers of 3 modulo 23:

• $3^1 = 3$
• $3^2 = 9$
• $3^{11} = 177147 \equiv 1 \pmod{23}$

Thus, the order of 3 modulo 23 is also 11.

Part (b): Finding the Order of 2 and 7 Modulo 27

Since 27 is not prime, Euler's function gives $\phi(27) = 27 \times (1 - \frac{1}{3}) = 18$. So, the order of any integer modulo 27 must divide $18$.

Step 1: Find the order of 2 modulo 27.

We need to find the smallest $k$ such that:

$2^k \equiv 1 \pmod{27}$

The divisors of 18 are $1, 2, 3, 6, 9, 18$. We check the powers of 2 modulo 27:

• $2^1 = 2$
• $2^2 = 4$
• $2^3 = 8$
• $2^6 = 64 \equiv 10 \pmod{27}$
• $2^9 = 512 \equiv 26 \equiv -1 \pmod{27}$
• $2^{18} = (-1)^2 = 1$

Thus, the order of 2 modulo 27 is 18.

Step 2: Find the order of 7 modulo 27.

We now look for the smallest $k$ such that:

$7^k \equiv 1 \pmod{27}$

Checking powers of 7 modulo 27:

• $7^1 = 7$
• $7^2 = 49 \equiv 22 \pmod{27}$
• $7^3 = 7 \times 22 = 154 \equiv 19 \pmod{27}$
• $7^6 = 19^2 = 361 \equiv 10 \pmod{27}$
• $7^9 = 10 \times 19 = 190 \equiv 1 \pmod{27}$

Thus, the order of 7 modulo 27 is 9.

Final Results:

• The order of 2 modulo 23 is 11.
• The order of 3 modulo 23 is 11.
• The order of 2 modulo 27 is 18.
• The order of 7 modulo 27 is 9.

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Would you like further details on any of these steps? Here are some related questions to explore:

1. How does Euler's theorem help in finding the order of an integer?
2. What are the properties of Euler's totient function $\phi(n)$ for composite numbers?
3. Can the order of any integer modulo $n$ exceed $\phi(n)$?
4. How do powers of integers behave modulo prime versus composite numbers?
5. How do divisors of $\phi(n)$ restrict the possible orders of integers modulo $n$?

Tip: Checking powers of integers modulo $n$ is easier by systematically testing exponents that divide $\phi(n)$.