15 years ago
Anonymous
Favorite Answer
Statement a is not true. For a counterexample:
1² (mod 6) = 1 and 5² (mod 6) = 1. However, 1 (mod 6) = 1 and 5 (mod 6) = 5.
Statement b is true. Here is the proof:
Given a² (mod p) = b² (mod p) for p prime. Then:
a² (mod p) - b² (mod p) = 0
(a² - b²)(mod p) = 0
((a+b)(a - b))(mod p) = 0
Therefore, p is divisible by (a+b)(a-b). Since p is prime, p must be divisible by either (a+b) or (a-b). That leaves two possibilities:
• If p = a + b, then a (mod p) = b (mod p).
• If p = a - b, then a (mod p) = -b (mod p).
That completes the proof.
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Oops, maybe b isn't true after all. For example, 3² (mod 7) = 2 and 4² (mod 7) = 2. However, 3 (mod 7) ≠ 4 (mod 7). I think this will be true whenever a + b = p.
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