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

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

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

A: Given, 3+9+15+..+(6n-3)=3n2

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: Prove that 3"< n! whenever n is a positive integer greater than 6.

A: Here we are to prove 3n&lt;n ! whenever n is a positiveinteger greater than 6.To solve this problem…

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

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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: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

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: Prove that the statement is true for every positive integer n.n &lt; 2n

A: n&lt;2n Let us prove this statement by using the Principle of Mathematical Induction. When n=1,…

Q: Prove or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

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: Prove that if n is a positive integer, then 133 divides 11"+1 + 122n-1

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Q: Prove that for any natural number N, there exists N consecutive integers none of which is a power of…

A: Please see the attachment

Q: 24. Show that if p is a prime and k is an integer such that 1sksp-1. then p divides ().

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

Q: Number theory.

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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 that 8 | (54" +7) for every positive integer n.

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

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Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

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Q: Prove: n! > 2"- for any positive integer n that is greater than 2.

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Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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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: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

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: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: How to solve this?

A: According to the given information, let p be a prime number such that p&gt;2. It is required to…

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

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

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

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

A: Given , n is an integer .To prove → if 3n+2 is even , then n is even using a proof by composition…

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

A: CASE I When n is even Let n=2m,m∈N⇒n3−n=2m3−2m⇒2m4m2−1=2η,where η=m4m2−1Thus,n3−n is even when n is…

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

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Q: Prove that for any prime positive integers P,VP is an irrational number

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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 that a positive integer n is p

A: Topic :- Number system

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 any integer greater than or equal to 48 can be represented as the sum of nonnegative…

A: Principle of mathematical induction: For each and every natural number n, mathematical induction is…

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

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

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Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: Prove that if p is a prime integer, then 3√p is irrational.

A: To Prove that: If p is a prime integer, then 3√p is irrational.

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

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Q: Prove that for all primes p that if p does not divide n2, then p does not divide n.

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

A: 13+23+33+---+n3=nn+122 Let us use the Principle of Mathematical Induction to prove the given…

Q: Prove that 30 divides n° – n for every nonnegative integer n. -

A: The given question can be solved as shown in step2.

Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

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

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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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