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 1 year 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....

Prove or disprove the following statements. (a) There is a simple graph with 6 vertices with...

Prove or disprove the following statements. (a) There is a simple graph with 6 vertices with degree sequence (3, 3, 5, 5, 5, 5)? (b) There is a simple graph with 6 vertices with degree sequence (2, 3, 3, 4, 5, 5)?

Prove that there exists a negative number.

Prove or disprove the following statements: a) If both x2 and x3 are rational, then so...

Prove or disprove the following statements: a) If both x2 and x3 are rational, then so is x. b) If both x2 and x3 are irrational, then so is x. c) If both x+y and xy are rational, then so are x and y.

Prove or disprove whether the function f: Z x Z -> Z x Z given by...

Prove or disprove whether the function f: Z x Z -> Z x Z given by f(x,y) = (2x+y, 3x-6y) is injective, surjective or both.

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...

L= {w belongs to {0,1}* | the number of 0's is a prime number} . Prove that the language L is not regular.

Use the method of exhaustion to prove the following statement: “For every prime number p between...

Use the method of exhaustion to prove the following statement: “For every prime number p between 30 and 58, 10 does not divide p − 9.” Prove that 0.17461461 . . . is rational (digits 461 in the fractional part are periodically repeated forever).

Prove or Disprove: that Zxmn is isomorphic to Zxm x Zxn if gcd (n, m) = 1

2.31. Show that for each of the following values of a and b, there exists x,...

2.31. Show that for each of the following values of a and b, there exists x, y in Z satisfying ax + by = 11. (i) a = 11, b = 0, (ii) a = 22, b = 11, (iii) a = 33, b = 22, (iv) a = 451, b = 33, (v) a = 484, b = 451. 2.39. Prove that gcd(ad, bd) = |d|gcd(a, b). 2.44. Does the Diophantine equation 12x + 33y = 1 have an...

Question