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

Prove or disprove each of the following statements.

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

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

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

(d) There exist 100 consecutive composite integers.

Expert Solution

Colby Messinger answered 2 years ago

Prove or disprove the statements: (a) If x is a real number such that |x +...

Prove or disprove the statements: (a) If x is a real number such that |x + 2| + |x| ≤ 1, then x 2 + 2x − 1 ≤ 2. (b) If x is a real number such that |x + 2| + |x| ≤ 2, then x 2 + 2x − 1 ≤ 2. (c) If x is a real number such that |x + 2| + |x| ≤ 3, then x 2 + 2x − 1 ≤ 2....

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In each case below either prove that the statement is True or disprove it by giving...

In each case below either prove that the statement is True or disprove it by giving an example showing that it is False. (i) If B is a fixed 2 × 2 matrix, then the set U = {A|A ∈ M22, AB = 0} is a subspace of M22. T or F (ii) If u, v, w are vectors in a vector space V , then span {u, v, w} = span {u + v, u + w, v +...

L= {w belongs to {0,1}* | the number of 0's is a prime number} . Prove...

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