5. Use direct proof technique to prove the following theorem: "If n is an odd integer, then n³ + 1 is an even integer."

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: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

A:

Q: Show that for all positive integers n, 13 + 2° + ...+ n³ = (1+2+3+ ...+ n)?

A:

Q: Prove that if n is an integer and n3 + 5 is odd, then n is even

A:

Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

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

A:

Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

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: 5. Show that 2 32 3" 31 3" for all positive integers n.

A:

Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

A:

Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

A:

Q: 3. Prove that if n is an odd integer then 7n+5 is even.

A:

Q: 7) Use a direct proof to show that: a) The sum of an even integer and an odd integer is odd. b) If n…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

Q: Prove that n- + 1> 2" when n is a positive integer with 1 <n<4.

A:

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 m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: 2. What are the two largest positive integers n satisfying n - 1 | n²-A?

A:

Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: 2. Possible whenn A- Zero b. Negutive integer C- Negative Yational d- Positive integer

A: We know that Laplace transform of

Q: 4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ +…

A: To prove that cube of any integer can be written as the difference of two sqaures.we have given hint…

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

A:

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

A:

Q: Prove that 3 divides n3+2n whenever n is a positive integer

A: We use mathematical induction. Let P(n) be "3 divides n3+2n" For n=1 n3+2n=(1)3+2(1)=3 We then note…

Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: show that no integer of the form n^3+1 is a prime other than 2

A:

Q: If nis a positive integer 1 then prove (D - a) n!

A:

Q: 2. Prove that if m and n are integers and mn is even, then m is even or n is even.

A:

Q: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

A:

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

A:

Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: - Prove that if n is an integer, then n = +. (Hint: use the proof by cases with respect to even and…

A:

Q: Prove that n² + 1> 2" when n is a positive integer with 1<n<4.

A: Given: is a positive integer with. To prove: .

Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

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: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

A:

Q: (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < a +b < b. (b) Prove by contradiction : For…

A:

Q: 9. Show that if n is a positive integer then n Σ k² ( x ) = n(n + 1)2²−² k k=0

A: Concept: Mathematical induction: Mathematical induction is a technique for proving that a statement,…

Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

A:

Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27

Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

A: