Prove or disprove the following statements a) Let a N such that a +1 is a prime number. Then a is even. b) Let p be a prime number, and a c N. Suppose pla³. Then pla.

Q: If a is an odd integer show that.

A: Given :-  a is odd integer  a-2 = 1    =>  a-2 = 1  =>  1a-2=11  =>  a2=1  =>  a=±1

Q: Prove the following: If m is an even number and n is an odd number, then n 2m is odd. Hint: start…

A: We prove that if m is an even number and n is an odd number , then n-2m is odd .

Q: The following table shows the solution of r + 4r - 12 = 0 by using some mumerical method. 1.75…

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Q: two consecutive odd numbers are both primes, they are called twin primes. One example of a pair of…

A: A) 19 and 21 : 21 = 7*3 so it's not prime. B) 39 and 41 : 39=13*3 so it's not prime. C) 49 and 51 :…

Q: Prove that if 7 is a prime number, then √ is an irrational number.

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Q: A statement S, about the positive integers is gi Sn: 11 +33 +55+ ... + (22n - 11) = 11n²

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Q: Find the greatest common divisor (a, b) and integers m and n such that (a, b) = am + bn. a = 65, b…

A: a) The objective of the question is to find the gcd of two numbers and also to represent the gcd as…

Q: Which of the following is not true? a) for any prime p, Jp is irrational. b) if Va is rational, then…

A: a.) To prove: For any prime p, p is irrational number. Proof: Let us proof this result by…

Q: Part 4. Prove the following mathematical statements. Let n be any integer. a. If n – 1 is odd, then…

A: a. It is given that n-1 is odd.

Q: Prove by induction that for sets A1, A2, . .,An, n A¡ i=1

A: For induction method we take a statement. We will show that the statement is true for n=1, n=2 then…

Q: Let p E Z Z be a prime number such that p > 2. Let n = 2p. Find all positive %3D divisors of n. NO…

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Q: d.) Find the second perfect number using the formula (2P-1)(2P – 1)and the next prime number for P.…

A: As per our guidelines we are supposed to answer only first three subparts of the problem. For the…

Q: Find UA, and ₁4, where: a. A₁ = {iº, i¹, i²) for every positive integer i. b. A₂ = {2i+1} for every…

A: a. ⋃i=1∞​Ai​ ​ is the set of all non-negative integers.b. ⋂i=1∞​Ai​={1}.Explanation:Let's find the…

Q: Prove or disprove the following statements: a. √√3 is irrational. b. There exists three consecutive…

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Q: 2. Prove the statement below using the correct proof method: a) using proof by cases, prove that…

A: This is a problem of finite math, logic and counting.

Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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Q: solve it math A, B,and C

A: Answer 1a)OR {x ∈ Z+|x≡1 (mod 8)}Answer 1b)Answer 1c)

Q: Find U14 and 14, where: a. A₁ = {{0,1,2) for every positive integer i. b. A₁ = {2i + 1} for every…

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Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

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Q: If p is prime and pl ab then p la and plb Select one: O True O False

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Q: Which of the following numbers is a perfect square or a perfect cube, but not both? 12 544 531 441 O…

A: considering option wise1) 1254412544 =112 , perfect square2) 531441531441=729 , perfect square

Q: Question If 0 < 0 < 2T and –12cos²0 – 8 = –17, determine the values of 0. If there are multiple…

A: The value of cos in the fourth quadrant is positive

Q: For any nonzero integers a, b, c, k. One of the following statements is true A. gcd(ka, kb) =…

A: Here only two options are given to choose. Option B) and C). Also option A), D) not for choosen.

Q: Write a proof of the following statement: if m and n are odd integers, then mn + 1 is an even…

A: The proof of given statement is given by direct proof method as follows :

Q: Give a direct proof of the claim: V n E N, Vk E N, if the last k digits ofn are divisible by 2k then…

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Q: Let n be a real number. Prove that if n is irrational, then 5n - 1 is also irrational.

A: Given that n is real number. If n is irrational number then we need to prove that 5n-1 is also a…

Q: Prove the given statement: The sum of an integer and its square is even.

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Q: If n is a perfect number, prove Σ d|n 1 2.

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Q: Construct a sieve of Eratosthenes for the numbers from 2 to 100. (Write all the integers from 2 to…

A: The given numbers,(a) (b) (c) (d) The objective is to determine whether the numbers are prime or…

Q: By dragging statements from the left column to the right column below, construct a valide proof of…

A: Given: For all integers n, if 19n is even, then n is even. To construct: Valid proof of the…