a) Prove that if n is an even integer, then n2 + 5n+2 is even. (Use a direct proof.) b) Prove that if x is an irrational number, then 5x +12 is irrational. (Use a proof by either contraposition or contradiction.)

Q: 3. Prove the statement "6 divides n³ - n for all integers n> 0".

A: We need to prove that 6 divides n3-n for all integers n≥0.

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

A:

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

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Q: If n is an even postive integer, then 3n+2^n-2 is odd.

A: Number problem....

Q: Use a direct proof to show that if a and b are positive integers, then ab+ba22.

A: Answer

Q: Take any arbitrary integer n and then square it, which one is true? n2 cannot be in the form 7k + 5.…

A: Solution is given as follows :

Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contradiction.

A: Given: 3n + 2 is an even integer Let n is not even, thus n is odd. Therefore, by using the…

Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

A:

Q: 1. Prove that the square root of 2 is not a rational number.

A: Definition of rational number : If x is a rational number then it can be written in the form of pq…

Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

A:

Q: Give a contrapositive proof of the following: If n is an odd integer, then the equation x2+ x -n=0…

A:

Q: Prove that if n is an integer, then 3n? + n + 14 is even

A: Case I - Let n be an odd integer , then n= 2k+1 where k is an integer. Then 3n2+n+14   =…

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

A:

Q: a and c are odd integers.  ab +bc is even for every integer b use indirect and direct proofs

A: Direct proof:- Given that, a and c are odd integers. a=2m+1, c=2n+1 Now,…

Q: Prove that the statement is true for every positive integer n.

A: Given that, 2 is a factor of n2+n   We have to prove this statement is true for every positive…

Q: Prove that √2 + √7 is an irrational number

A:

Q: p + 2 is a prime number, prove that p2+3 is divisible by 4. (2.5

A: Prime and divisibility

Q: 1. If today is Saturday, what day of the week will it be 1,500 days from today? 2. Prove that 4/2 +…

A: The explanation of the solution as follows:

Q: Which statement is not correct? Sum of two irrational numbers is irrational. O For every integer n,…

A: The incorrect statement is : sum of two irrational numbers is irrational.   the detailed solution is…

Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

A: We have to factories the equation  x3+3x2+2x-258,474,216=0

Q: Use contradiction to prove:  if 3n + 2 is an odd integer, then n is odd.

A: consider 3n+2 as odd integer  and let  n is even  therefore, n=2k for some integer k therefore,…

Q: (4) Consider the statement "There exists an even integer n, such that for all rational number q, we…

A: Statement "There exists an even integer n, such that for all rational number q, we have n²>q"

Q: Prove 3√3  is irrational

A: See the attachment.

Q: Prove that a positive integer n is prime number.if no prime p less than or equal to n divides n.

A: Let us assume that the contradiction of the statement , let  n ≥ 0 be a composite number .  Thus , n…

Q: Use indirect proof to show that if n is an integer and n^3 + 5 is odd, then n is even.

A:

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

A:

Q: prove that √2+√3 is irrational

A: For rational we needed form p by q P and q are integer

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

A:

Q: Prove by using proof by contrapositive:   Prove that if 3n + 2 is odd, then n is odd.

A:

Q: Do you think it is worthwhile, or useful, to solve the puzzles proposed in Euclid's theorem of…

A: The objective is to solve the puzzles proposed in Euclid's theorem of primes There are infinitely…

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

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Q: Use a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

A:

Q: (2) (a) Show , if n2 - 6n + 5 is even, then n is odd for every integer n. Prove by contrapositive.…

A: Since you have asked multiple question, we will solve any one question for you. If you want any…

Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Use proof by contradiction to show that for all integers m, 7m+4 is not divisible by 7.

A: Recall the following fact about divisibility.

Q: Prove that 2+/3 is irrational

A: Given,          To Prove : 2+3 is irrational.

Q: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: Given: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even

Q: Show that sqrt(3) is an irrational number. (And show and use that if 3 divides n^2, then 3 divides…

A:

Q: Using the parity theorem and contradiction, prove that for any odd positive integer p,√2p is…

A:

Q: Use proof by contradiction to show that the sum of two odd integers(k and l) is eve

A: An even number is a number that is divisible by two. All even numbers finish with one of the digits…

Q: Give a contrapositive proof of the theorem  If n is an integer and 3n+2 is even ,then n is even

A: We have to prove given statement by contrapositive proof.

Q: Use a direct proof to show that if a and b are positive integers, then ab+ba >=2

A: Claim:  Where a and b are positive integers. Since commutative property holds in set of integers…

Q: Prove that cube root of 2 is irrational by contradiction

A:

Q: For all prime numbers  a, b, and c, (a^2 +b^2) does not equal c^2. prove by contradiction

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.