How do I prove: If n > 1 is a natural number then there is a prime number p such that p|n.

Q: 3. If 2" – 1 is prime, then n is prime. 4. Let x be a real number. If x > 0, then > 0.

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Q: How many digits does the smallest positive integer K have, while satisfying all of the following…

A: We will construct the number K bu fulfilling the conditions from bottom to top.

Q: Show that if n is odd and 3 n, then n2 = 1 (mod 24).

A: The solution is below.

Q: A prime number p is called special if there exist primes p, P2, P3, P4 such that p = pP + P2 P3 -P4…

A: Prime number problem

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: 6. A classic unsolved problem in number theory asks if there are infinitely many pairs of "twin…

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Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: Using direct proof method, prove the following. If n is an even integer then n² + 11 is odd. Answer:

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Q: For which natural numbers n is n3-10n2+28n-19 a prime number

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Q: 3. Suppose a formula that you need to prove by mathematical induction is true for all the natural…

A: Given problem:- Suppose a formulas that you need to prove by mathematical induction is true for all…

Q: 1. Using proof by contradiction, prove Let n be an integer. If n- is even, then n is even.

A: Given: Let n be the integer.  To prove : If n2 is even, then n is even.

Q: 9. This question is a little more challenging. How many integers in the interval [1, 99 999] (i.e.,…

A: Given data:   The given sum of all digits is k=10. The number of digits that can be used is n=5.…

Q: 7. If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

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Q: Prove or disprove: for every natural number n,2+5+8+...+(3n-1)=n(3n+1)/2

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Q: 2. Using Direct proof, Prove: if m is an even integer and n is an odd integer, then mn is also an…

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Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p, at least one of…

A: GIVEN: Prove that there are no 3 consecutive odd prime numbers

Q: Let h be a prime number and let k be some integer. What are the possible values of ged(h+k, h-k)?

A: We have given that ,  h be any prime number and k be some integer. We need to find possible values…

Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: Using direct proof method, prove the following. If n is an even integer then n²+ 13 is odd.

A: n is even, so take n=2k and put on the expression.

Q: 8. * ) Prove for any odd integer n, we have Proof: Let n be an odd integer. We want to show finish…

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Q: f we want to prove by contradiction that there is an infinite number of prime numbers, we must…

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Q: Consider the statement: for all integers n, if n is even then 8n is even. Prove the statement.…

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Q: What is the converse of the statement "If n is an even integer, then 5n - 3 is an odd integer."?

A: A statement which is  obtained by reversing the given hypothesis and conclusion of a conditional…

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: Show that if n is an integer and n³ + 5 is odd, then n even using a) a proof by contraposition. b) a…

A: Proofs: The proof is a valid argument that establishes the truth of a statement. Types of Proofs:…

Q: 3. Show that if m and n are integers and n +5 is odd then n is even using a) a proof by…

A: We have to show that if m and n are integers and n3+5 is odd then n is even using (a). A proof by…

Q: Part III: Prove or Disprove the following 9. For every positive integer n, n2 – 2 is not prime is…

A: To prove:- For every positive integer n, n² - 2 is not prime is not necessary for n to be odd.

Q: Using direct proof method, prove the following. If n is an even integer then n2 + 13 is odd.

A: Direct proof

Q: How many consecutive zeroes are there at the end of 100! (100 factorial). For example, 12! -…

A: How many consecutive zeroes are there at the end of 100! We know that…

Q: how many pairs of primes p and q are there such that q-p=3? Prove your answer.

A: Here, we need to find all the possible pairs of prime numbers p and q such that the difference (q-p)…

Q: How do I prove: If n is a natural number and d|n prove that d ≤ n.

A: How do I prove: If n is a natural number and d|n prove that d ≤ n.

Q: 9. (a) Show that if n is an integer, then: -o3* (") = 4". %3D (b) Let n be a positive integer. Show…

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Q: Prove or disprove: let n be an integer. if 3 divides n^2, then 3 divides n.

A: Let us consider some cases for proving that n be an integer. if 3 divides n^2, then 3 divides n.For…

Q: (Proof) For any integer n, n² + n is even.

A: We know the product of consecutive integers in a even integer.

Q: Let p be a prime number, n 2 2. Show that 6(pn) = po(n) if and only if p n. %3D

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Q: Prove or disprove the validity of the following statement: For any natural number n, n2 + n is…

A: We have to prove or disprove the given statement. The given statement is for any natural number n,…

Q: Using direct proof method, prove the following. If n is an odd integer then n2 + 13 is even.

A: We have to prove that, If n is an odd integer, then n2+13 is even. Since n is odd, we can consider:…

Q: 3. Let h be a prime number and let k be some integer. What are the possible values of gcd(h + k, h -…

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Q: Prove that if p> 3 is a prime number, then 24 | (p2 - 1).

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Q: Find all natural numbers n that satisfy two of the given statements 1) Number n is a perfect square.…

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Q: For which natural numbers n is n^3-10n^2+28n-19 a pri For which natural numbers n is…

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Q: Which statement is universal? - 5 is not a multiple of 10 - 2n + 1 is prime for some n ∈ N - n…

A: We have to solve given Questions.

Q: (a) Prove the following statement about natural numbers from the axioms N1 to N13. Apart from the…

A: Given:  a<b and b<c, ∀a, b, c∈ℕ To prove : a+1<c

Q: 1. Using proof by contradiction, prove Let n be an integer. If n' is even, then n is even.

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Q: Using direct proof method, prove the following. If n is an even integer then n2 + 13 is odd. Answer:

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Q: Prove the statements: Let n be an odd positive integer. Then the sum of n consecutive integers is…

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Q: Prove that the sum of the first n natural numbers is given by this formula: n(n + 1) 1+2 +3 +. ..…

A: As per our guidelines, we are supposed to answer only one question. Kindly repost other questions as…

Q: (a) Give an example of a natural number n such that n is composite, even, and √n is not a rational…

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Q: 3- if a belongs to the set of all even natural numbers, and b belongs to the set of natural numbers…

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