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Prove or disprove each of the following statements. (a) There exists a prime number x such...

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Advance Math

Prove or disprove each of the following statements.

(a) There exists a prime number x such that x + 16 and x + 32are also prime numbers.

(b) ∀a, b, c, m ∈ Z +, if a ≡ b (mod m), then c a ≡ c b (modm).

(c) For any positive odd integer n, 3|n or n 2 ≡ 1 (mod 12).

(d) There exist 100 consecutive composite integers.

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