Question

show that every odd prime is congruent to 1 or 3 modulo 4. if p > 3 is prime, to what can p be congruent modulo 6?

Answer 1

Every odd prime is congruent to 1 or 3 modulo 4. When p > 3 is prime, p can be congruent to 1, 3, or 5 modulo 6.

Step-by-step explanation:

An odd prime number is a prime number that is not divisible by 2. We want to show that every odd prime is congruent to 1 or 3 modulo 4. Let's assume that p is an odd prime. Since p is odd, we can write it as p = 2k + 1, where k is an integer. Now, we can check the possible remainders of p when divided by 4:

1. If k is even, then p = 2(2n) + 1 = 4n + 1, which is congruent to 1 modulo 4.
2. If k is odd, then p = 2(2n + 1) + 1 = 4n + 3, which is congruent to 3 modulo 4.

Therefore, every odd prime number is congruent to either 1 or 3 modulo 4.

If p > 3 is prime and we want to determine the possible congruences of p modulo 6, we can use a similar approach. We can write p as p = 6n + k, where n is an integer and k is the remainder when p is divided by 6. Since p is odd, we know that k must be odd. Therefore, the possible values for k are 1, 3, or 5. This means that p can be congruent to 1, 3, or 5 modulo 6.