Question

Show that if n is an integer then n² ≠¡ 0 (mod 4) or n² ≠¡ 1 (mod 4)?

Answer 1

To show that if n is an integer, then n² ≠ 0 (mod 4) or n² ≠ 1 (mod 4), we can use the concept of modular arithmetic. If n² is congruent to 0 or 1 modulo 4, it means that n² leaves a remainder of 0 or 1 when divided by 4. By considering the two cases of n² ≡ 0 (mod 4) and n² ≡ 1 (mod 4), we can show that n² is not congruent to 0 modulo 4 or 1 modulo 4.

Step-by-step explanation:

To show that if n is an integer, then n² ≠ 0 (mod 4) or n² ≠ 1 (mod 4), we can use the concept of modular arithmetic. In modular arithmetic, we consider the remainder when a number is divided by another number. If n² is congruent to 0 or 1 modulo 4, it means that n² leaves a remainder of 0 or 1 when divided by 4.

Let's consider the two cases:

1. If n² ≡ 0 (mod 4), then n² is a multiple of 4. This means that n is also a multiple of 4. If n is a multiple of 4, then n² is a multiple of 16 (n² = 16k, where k is an integer). However, this contradicts the fact that n² is congruent to 0 or 1 modulo 4.
2. If n² ≡ 1 (mod 4), then n² leaves a remainder of 1 when divided by 4. In this case, n could be an odd or even number. Let's consider both possibilities:

• If n is an odd number, then n = 2m + 1 for some integer m. Substituting this into n² = (2m + 1)², we have n² = 4m² + 4m + 1 ≡ 1 (mod 4). Therefore, n² is congruent to 1 modulo 4.
• If n is an even number, then n = 2m for some integer m. Substituting this into n² = (2m)², we have n² = 4m² ≡ 0 (mod 4). Therefore, n² is congruent to 0 modulo 4.

In both cases, n² is either congruent to 0 modulo 4 or congruent to 1 modulo 4, which means that n² ≠ 0 (mod 4) or n² ≠ 1 (mod 4).