5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form 4m +1.

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: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

A:

Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

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

A: 3n+2 and 5n+3 are relatively prime for every integer n.

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

A:

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: Prove that n! > 2" for n > 4.

A:

Q: п 12. Show that Ë O)=)2"-". Using Combinatorial Proof. k=0

A: To prove: ∑k=0nnkkm=nm2n-m

Q: Show that n!> n3 if n is large enough.

A:

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

A:

Q: x4 >is ........in Q[x] Not prime ideal ONone of the choices O Prime not maximal

A: Answer is maximal ideal , i.e , last option , and proof is given below.

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: Find the order of the residue class 5 in Z/29Z.

A:

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

A: Take k=4 and C=6

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: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

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…

A:

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: Show that Ekin d(k)3 = (Ekn d(k))2 for all positive integer n. %3D

A:

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

A:

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

A:

Q: We need to attempt to find an exact value of n such that  P(r ≥ 6) = 0.98. Clearly to guarantee…

A: We have to fill blanks from given statement Clearly to guarantee at least six people are literate,…

Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Show that yn for all n=1, 2₁- ✓

A:

Q: Prove T(n)= 2T(n-1)+n can be in 0(n2")using represented telescoping method

A:

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: 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: Show that 2n+1≤2^n  for all n≥3.

A:

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

A: Use binomial expansion:-

Q: 5. Prove that for all n n [j³ = j=1 n² (n + 1)² 4

A:

Q: Number theory

A: Consider the provided question,

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

A:

Q: Show that φ(n2) = nφ(n) for every n ∈ Z+.

A:

Q: Use any method to deter (a) 13 The series 8.

A:

Q: Let N=20. Show using the Sieve of Eratosthenes how to find all prime numbers between 10N and 10N+100…

A:

Q: (12) What values can be assigned to r so that {4n+r} will include all odd primes?

A:

Q: 2. Show that for every n 2 2 and every 0 <k<n- 2, within the triangle of Binomial Coefficients k+2…

A: We will use properties of binomial coefficient  nCr  +  nCr-1=n+1Cr+1Note : nCr=nr=n!r! n-r!   Proof…

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: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

A:

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

A:

Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: Prove E(2k – 1) = n² for n > 1. - k=1

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 6. Show that (") = 3() + 6n (5) +n³. Using Combinatorial Proof.

A:

Q: 7. Suppose a1, a2,...,a, are integers, r is a solution to the equation x" + a1x"-1 + a2x"-2 +... +…

A: for question number 7  we use contradiction to prove the statement.

Q: 3. Prove that, Σk() = n2n-¹. k=1

A: