Problem 1. (1) What are the squares modulo 4? (2) Let p be an odd prime and assume that p= a²+t for some a, be Z. Show that p=1 (mod 4). (3) Show that if p is an odd prime satisfying p = 3 (mod 4), then p is irreducible in Z[i).

Q: Question 2. Prove that 2"32n – 1 is divisible by 17 for all n E N. Hint. Induction. -

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Q: Question 5: For the recurrence relation Xn+2 – 3xn+1+2xn = 1 where xo = 0 and x1 = 1, (a) List the…

A: It is given that

Q: (d) Given that n is even, how many 00-free bitstring of length n where n is even have exactly (끌 +…

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Q: Find a solution x E Z, 0 < x < n, for each of the congruences ax = b (mod n) Note that in each case,…

A: The given congruence relation is 25x=31mod 7. Compare 25x=31mod 7 with ax=bmod n and obtain the…

Q: 8. Find the remainder of x2021 + x2020 + 1 modulo x2 – 1 in Q[x].

A: Remainder of the Polynomial

Q: Problem 5. Consider the recurrence T(n)= T(n-1) + (n). Prove that T(n) is (n²).

A: Given is the recurrence T(n)=T(n-1)+θ(n) The objective is to prove that T(n) is θ(n2).

Q: Problem 1. Prove by induction that, for every n 2 1, п(n + 1) El-14? = (-1)*. 2 i=1

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Q: Question 6.7. Prove that x'(Km) = n if n is odd and x'(Km) = n – 1 if n is even without using…

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Q: Problem 4: Prove that for all nonnegative integers n, n³ – 7n + 15 is divisible by 3.

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Q: Problem 4: a) Use the strong form of the Sterling approximation to compute the probability of…

A: a)   The strong form of Sterling’s approximation is, The probability of getting exactly 500 heads…

Q: Suppose n E Z. Prove that if n is even, then n2 + 5n + 4 is even.

A: Note: We know that the square of an even number is always an even number, since  Let x is an even…

Q: Problem one : Prove or disprove the following statements: (a) If n e Z then the equation n' + (n +…

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Q: Problem 5.5 Prove that for every positive integer n, 1 1+ V2' V3 1 +...+ Vn 1 > 2(Vn+1- 1)

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Q: QUESTION 9 Use Mathematical induction to prove that 2 divides n2+5n, n21.

A: Question 9: Let n=1 is true for Let the result be true for n=k,  We need to prove the result is…

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

A: Take k=4 and C=6

Q: Problem 1.3: proof by induction Prove that %3D k(k + 1) n+1 for n EN and n> 0.

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Q: Problem 5.2. Prove II 1 for all n> 2 by the generalized PMI. 2n i=2 Hint. Follow the steps of the…

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Q: PROBLEM 8 (2) Define P(n) to be the assertion that: n(n + 1)(2n + 1) j=1 (a) Verify that P(3) is…

A: Given: ∑j=1nj2=nn+12n+16

Q: Suppose that phi (x) belongs to Aut (Zn) and a is relatively prime to n.  If phi (a)=b, determine a…

A: According to the given information it is given that:

Q: Express 1z and 2z in exponential form.  2) Find modulus, |z1 z2|, and argument, arg(z1 z2 ).  need…

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Q: Problem 9. Prove that every amount of postage of 12 cents or more can be formed using just 4-cent…

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Q: Problem 5.8 Prove, for all positive integers n, 1 3 2n – 1 1 < V3n 2 4 2n

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Q: Problem 12 (a) Prove that if m is an integer of the form 2k, for some k 2 0, then m is divisible by…

A: Euler's Totient function:  ϕp=p-1  ϕpa=pa-1p-1  and  ϕmn=ϕm ϕn  when gcd m,n=1

Q: Prove by induction that, for every positive integer n, we have Xn i=1 (6i + 1) = n · (3n + 4)

A: Hey there. I am solving part (b) Do drop a like if this solution helps you out.

Q: Question 3. Let n > 2 be an integer. Show that n is either a prime or it has а prime faclor < Vп.

A: Question: Let n be an integer such that n≥2. Prove that either the number is a prime number or it…

Q: Problem 1: Let p be any prime. Prove that (n² – 2)(n² – 7)(n² – 14) is divisible by p for infinitely…

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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: Problem 1: Prove that for every integer n > 7, (n – 2)! > n² . (Hint: using the requirement n > 7 as…

A: We use induction hypothesis to prove the result.

Q: Problem 5.5 Prove that for every positive integer n, 1 1+ V2 1 + V3 1 > 2(Vn +1- 1)

A: It is require to prove that 1+12+13+…+1n&gt;2n+1-1 for every positive n. Use mathematical induction…

Q: Problem 5.3 Prove that 5it = n(n+1)(2n + 1) (3n² + 3n – 1) 30 j=1 whenever n is a positive integer.

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Q: Theorem 4.10. Let p be an odd prime. Then among 1, 2, ,p-1, exactly (p-1)/2 many are quadratic…

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Q: Problem 4.2 Prove that for every positive integer n, n(n + 1)(n + 2) 1.2+2.3+ + n(n + 1) = ... 3

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Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: Problem 1. Let p be an odd prime. Prove that for all i with 1 < i sp - 2 one has p| 1' + 2' + ·.. +…

A: Given that p is an odd prime. We need to prove that for all i with 1≤i≤p-2,…

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

A: The given statement is,

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

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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: Please solve part b.

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Q: Use Theorem 1 to determine the number of integral solutions of the polynomial equation a^2 + b^2 =…

A: First of all, find the prime factorization of 23400 as:23400 = 23×32×52×13Since, here

Q: Problem 3. For any integers n, k,r where n > k > r > 0, give a combinatorial proof of the following…

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Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

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Q: Question 4 Use Mathematical Induction to show that 512|8 · 5" +6. 32n – 80n – 14 for all n E N.

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Q: In a Wilcoxon Mann Whitney test n1 is 9 and nprime is 7. What are the correct values for K1 and K2 ?…

A: In a Wilcoxon Mann Whitney test n1 is 9 and nprime is 7.  We need to find the correct values for K1…

Q: Problem 4.3 Prove that for every positive integer n, 뉴 1 > 2(Vn+1- 1) 1+

A: Consider the given statement as "For every positive integer n, 1+12+13+…+1n&gt;2n+1-1". Take the…

Q: Problem 3. Prove by induction that, for every n 2 1 (2i – 1) = n? i=1

A: According to our guidelines if there are multiple questions, we have to solve the first question…

Q: Please help me solve this abstract algebra problem with a better explanation. The MI stands for…

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: Theorem 11.3. If p and q = 2p+1 are primes, then either q | Mp or q |Mp +2, but presented next. not…

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Q: 6. Find a prime number p for which the equation x² = -1 mod p has no solution.

A: Given is a  modular equation.   To Find: A prime number p such that the given equation has no…

Q: In parts (a)- (e), involve the theorems of Fermat, Euler, Wilson, and the Euler Phi-function. (a)…

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Q: Question 13 Use Mathematical Induction to prove that for all integers n ≥ 1: (1 x 2) + (2 x 3) + (3…

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