Question

Suppose that 7x ≡ 28 (mod 42). by theorem 3.9, it follows that x ≡ 4 (mod 6). check this explicitly using theorem 3.6.

Answer 1

To check if x ≡ 4 (mod 6), we can use theorem 3.6 to simplify the equation. We find that x ≡ 4 (mod 6) is true.

Step-by-step explanation:

To check that x ≡ 4 (mod 6) we can use theorem 3.6, which states that if a ≡ b (mod n) and c ≡ d (mod n), then ac ≡ bd (mod n). In this case, we have 7x ≡ 28 (mod 42), and we want to show that x ≡ 4 (mod 6).

Let's apply theorem 3.6. Since 7 ≡ 1 (mod 6) and 28 ≡ 4 (mod 6), we have (7)(x) ≡ (1)(4) (mod 6), which simplifies to x ≡ 4 (mod 6).

Therefore, we have confirmed using theorem 3.6 that x ≡ 4 (mod 6).