6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

A: See the attachment

Q: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

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Q: find a Positive integern for which the equality holds %3D

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Q: 9. Show that there are infinitely many primes of the form 16k + 1. More generally, show that for any…

A:

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: rove that the negative of any even integer is even

A: Assume 2n to be an even integer. 2n = even integer -2n = negative of even integer  Let a = -2n be an…

Q: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

Q: 4. Find the coefficient of the term containing x' in the expansion of (x-2y)“.

A:

Q: 109. Every nth-degree polynomial has (n − 1) critical numbers.

A: For example consider the 3rd order polynomial off(x)=x3, and its derivative is f'(x)=3x2, sosolving…

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

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Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

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Q: Determine whether the statement is true or false. Justify your answer with a proof by Direct Method…

A: The product of any two integers is odd. - True

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: 1. Prove that the negative of any even integer is even. 2. Prove that the difference between an even…

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: Let n and k be positive integers with 2k + 1 ≤ n. Find the number of ways to choose k children from…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

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: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

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Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: Suppose that Hilbert’s Grand Hotel is fully occupied on the day the hotel expands to a second…

A: We know, Hilbert's Grand Hotel is a thought experiment which gives counterintuitive property of…

Q: 4. Use the binomial theorem to find the first th ree terms in ascending powers of x of (1

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Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

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Q: Ahmet borrows n TL from Burak. Starting from today, Ahmet pays back to Burak 2 TL or 3 TL at each…

A: Given that Ahmet borrows n TL from Burak. Starting from today, Ahmet pays back to Burak 2 TL or 3 TL…

Q: If n>1 prove the sum %3D is not an integer. 1无

A:

Q: Proceed as in Example 3 in Section 6.1 to rewrite the głven expression using a single power serles…

A: Given ∑n=2∞nn−1cnxn−2+∑n=0∞cnxn+2 We have to rewrite the given expression using a single power…

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: 4) Find the Term that free from x of the expansion (x2 +10

A: Use binomial expansion:-

Q: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: Obtain the expansion for (3 + 2x)* in ascending powers of x

A: According to binomial expansion,  a+bxn=an+n1an-1bx+n2an-2b2x2+n3an-3b3x3+....+nn-1abn-1xn-1+bnxn…

Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

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Q: s) Express the infinite contimod fraction [2, 1,5 quadratic irrational- as a ity.

A: We need to express the infinite continued fraction 2, 1, 5¯ as a quadratic irrationality. The…

Q: Find the first 4 terms in the expansion of (2 + x²)° in ascending powers of x.

A: Given expression: 2+x26  ……1   To find: First 4 terms (in increasing power of x) in the expansion of…

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: 3. Show that all the irrational numbers in the interval [0, 1] is dense there.

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Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

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Q: Find the ordinary generating function for the sequence 0,0,0,0,1,1,1...(Hint: it is not a…

A: We need to find an ordinary generating function. The sequence is 0,0,0,0,1,1,1,.... Consider:…

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: how that all the irrational numbers in the interval [0, 1] is dense there.

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Q: A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second day, he…

A: Given - A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second…

Q: 4. Prove that the L{2t²} = 4 S3

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Q: 3. Prove that 0 divides an integer a if and only if a 0.(In your proof. you can freely ILNE your…

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