1. **State the problem:** We are given a prime number $p$ such that $$p^2 \equiv 1.$$ We need to find which value of $p$ from the options satisfies this condition. 2. **Understand the congruence:** The congruence $$p^2 \equiv 1$$ means that $p^2 - 1$ is divisible by some modulus. Since no modulus is given, we interpret this as $p^2 = 1$ in integers, which is only true for $p = \pm 1$. But since $p$ is prime and positive, this is not possible. Likely, the problem means modulo some number, but since it's not specified, we test the options by calculating $p^2$ and checking if it equals 1 modulo $p$ or modulo some implied modulus. 3. **Check each option:** - For $p=2$: $2^2 = 4$, and $4 \equiv 1 \pmod{3}$? No clear modulus given. - For $p=5$: $5^2 = 25$, $25 \equiv 1 \pmod{24}$, but again no modulus. Since the problem is ambiguous, the best interpretation is to check if $p^2 - 1$ is divisible by $p$, which is always true since $p^2 - 1 = (p-1)(p+1)$ and $p$ divides $p^2$ but not 1. Alternatively, the problem likely means $p^2 \equiv 1 \pmod{p}$, which is always true. Given the options, the only prime $p$ such that $p^2 \equiv 1 \pmod{p}$ is trivially true for all primes. Hence, the problem likely means $p^2 \equiv 1 \pmod{p}$ or modulo some other number. Since the problem is ambiguous, we move to the second question. --- 1. **State the problem:** Given the sum of the first $n$ odd numbers $$S_n = 1 + 3 + 5 + \cdots + (2n - 1),$$ find the value of $$S_{n+1} - S_n.$$ 2. **Recall the formula:** The sum of the first $n$ odd numbers is known to be $$S_n = n^2.$$ This is a key formula. 3. **Calculate the difference:** $$S_{n+1} - S_n = (n+1)^2 - n^2 = (n^2 + 2n + 1) - n^2 = 2n + 1.$$ 4. **Interpretation:** The difference between the sum of the first $n+1$ odd numbers and the first $n$ odd numbers is the $(n+1)$th odd number, which is $2n + 1$. 5. **Answer:** The correct choice is A. $2n + 1$. **Final answer:** $S_{n+1} - S_n = 2n + 1$.