1. Let n be an integer. Prove that if n is even, then n2 + 2n + 3 is odd.

Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

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Q: Prove the following statement. 1) if k is a positive integer, then 3k+2 and k+1 are relatively…

A: We have to prove: If k is a positive integer, then 3k+2 and k+1 are relatively prime.

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

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Q: Prove that 6 divides n^3 − n whenever n is a non-negative integer

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

A: Number problem....

Q: Show that if n is an integer and 3n3 + n is odd then n is odd, by using a) Proof by Contrapositive…

A: To prove i - if n is integer and 3n3+n is odd then n is odd a) By contrapositive :   (P→9)≡(~q)…

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

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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: 1. If n is an integer, then 2n +1 is odd.

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Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

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Q: 1. Prove that nº + n is even for every integer n.

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

A: Topic:- algebra  Note :- if more than one question uploaded than only first question need to be…

Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: 1. Let n be an integer. Prove by contradiction that if n? + 5 is odd, then n is even.

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

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

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Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

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

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Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

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Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: Show that if n is an integer and 3n3 +n is odd then n is odd, by using a) Proof by Contrapositive b)…

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Q: 6.20. For which integers n > 2 does there exist a o e A, such that |o|> n?

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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: Let n be a positive integer. Prove that (:) n +2. +...+n. n2"-1.

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Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

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Q: 1 If n is a positive integer, then (D-a %3D n!

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Q: Let n be an integer. Prove that if 7 divides (n – 2) or 7 divides (n – 5), then 7 divides (n² + 3).

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

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

Q: 1. Prove: If n is odd then 5n + 4 is odd.

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Q: . Let n be an integer. Prove that n is odd if and only if 3n? – 7 is even.

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Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

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

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Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Prove that if 10|n, then 10|n^2+n

A: Prove that if 10|n, then 10|n^2+n

Q: 4. Prove that for every n > 1,3 divides n³ – n.

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

Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

Q: 11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

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: Let m, n, p be integers. Use direct proof to show that if m+n and n+p are odd then m+p is even.

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Q: 2. Prove that if 2n² + 7n is even and n is an integer, then n is even.

A: We have to prove that the given statement is true by the help of mathematical induction

Q: Prove that for every non-negative integer n the value of n^2+n+41 is prime

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Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

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

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Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

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Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

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