Recall that n is abundant if O(n) > 2n. Prove that if n = 2m-1(2m - 1) where m is a positive integer such that 2m - 1 is composite, then n is abundant.

Q: Show that the middle term in the expansion of (1 + x)2¹ is (1-3.5 (2n- -1)) 2"x", where n is a…

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Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: Suppose r is a non-negative real number and n is a natural number. Use the Binomial Theorem to prove…

A: The binomial theorem states that for the positive integer value n: x+yn=∑k=0nCknxk+yn-k

Q: If 2n +1 is prime for some n ≥ 1, prove that n is a power of 2.(Primes of the form 2n +1 are called…

A: Given: 2n+1 is prime for some n≥1. To Prove: n is a power of 2.

Q: The first term of the expansion of (j – 3n)a is –-3nª. True or False?

A: The \displaystyle \left(r+1\right)\text{th}(r+1)th term of the binomial expansion of \displaystyle…

Q: 1 1.3 1 Question 2. + + 3.5 1 5.7 + ... + 1 (2n-1) (2n + 1) for n ≥ 1

A: As per the question we are given the following series : 1/(1×3) + 1/(3×5) + 1/(5×7) + ... +…

Q: Let r be a real number and let k be an integer. We then define the binomial coefficient () by r(r-1)…

A: Given, r is a real number and k is an integer  The Binomial Coefficient rk by rk= r (r - 1) --- (r -…

Q: Identify the expression of x for which the following equation holds for every positive integers n…

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Q: Give a formula for o(n) as a product over the distinct primes dividing n.

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Q: "EZ¹, 2n + 3 is prime."

A: We have to prove or disprove the given statement using given information.Concept required:1) One…

Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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Q: Prove that o – 1 is divisible by 4.

A: Even numbers are numbers divisible by 2. Odd numbers are numbers that are not divisible by 2. The…

Q: Let k be a positive integer. Show that 1" + 2 + 3™ +....+(n – 1)m + n™is O (n™+1). -

A: According to the given information, it is required to show that:

Q: Find a Big-O estimate of (2n+n2)(n3+3n)

A: Given the given expression is 2n+n2n3+3n. Big-O notation: the fn is Ogn for positive constant C and…

Q: Let n be a positive integer and p a prime that divides n. Prove that pis prime in Zn.

A: It is given that, n be a positive integer and p be a prime that divides n.  We have to prove that, p…

Q: Icm(a,b) = ab if and only if a and b are relatively prime.

A: We know the relation between product of the two numbers their L.C.M and G.C.D Let a and b two…

Q: Show that for every positive

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Q: e 27 > 3n2 +3n +1 for all n E Z†, n > 8

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Q: . Use Mathematical Induction to show that for all n > 1. 71 i=1 F2i = F2n11-1

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Q: Fill in the blanks in the proof of the of fact that 3" – 1 is a multiple of 2 for every natural n.…

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Q: 1. Prove that for all positive integers n, n Σj2i = (n − 1)2n+¹ +2 j=1

A: To prove that,              ∑j=1nj2j=n-12n+1+2.

Q: Let n be a positive integer. Suppose that there are three pegs and on one of them n rings are…

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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: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

A: Composite numbers contain prime factors other than 1. Such as if a number has prime factors of 2, 3,…

Q: Show that 32n 1 is always a multiple of 8 for all n = N = {1, 2, 3,...}. We say one integer A is a…

A: Step 1:Step 2:Shows a proof by induction that 32n - 1 is always a multiple of 8 for all natural…

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: Let a and b be odd positive integers. If there exist integers s and t with as+bt=4 then gcd(a,b)=4 a…

A: We solve given problem with the help of Extended Euclidean Algorithm

Q: Prove by mathematical induction that for every positive integer n, 1 1 1 1 П 3i 3і Зі + 1 (Зп + 1)!…

A: Since you have posted multiple questions and we can answer one question at a time. So, answered…

Q: 4. Prove that for all n E N η Σ( r=0 = = (3) Hint: Examine the coefficient of x² by expanding both…

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Q: -1/2 p. Find the coefficient of x" in the expansion of (1– 4x)2.

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Q: .. (a) Prove by mathematical induction that 9**2 – 8n – 17 is divisible by 64 for all positive…

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Q: Use the binomial theorem to prove that n <-D₁ (17) 3¹-*. 2″ = 2-D² k=0

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