For any n> 1, let (n) be the number of positive integers a

Q: Let = e²i/n be an n-th root of unity. Show that (k(1-n) Resck (27²1) n for all 0 <k<n-1. =

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Q: e3x e(n+1)x = 1+ e" + 2 enx Sx 6. n! (n + 1)!

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Q: Show that x4 + 1 is reducible over Zp for every prime p.

A: In this question, we have to show that x4+1 is reducible over ℤp for every prime p.

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: For a non negative whole number n, let the function fn be given by (picture). where n!= 1*2...n,…

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Q: Prove that for any prime p and positive integer n, (p") = p" – p"-1.

A: Number theory, Euler totient function property

Q: n be a positive integer. (a) Prove that if 0 < a < b, then a" < b". HINT: Use mathematical…

A: To prove a statement using mathematical induction, we first prove it for n=1 and n=2. Then assume…

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: 2. Let P(x) = ax² + bx + c be an irreducible polynomial in Q[x). Prove that (a) P(x) has two…

A: Let P(x)=ax2+bx+c be an irreducible polynomial in Q[x] a) To show P(x) has two distinct roots, let…

Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

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Q: Given that f(n) is a function for all non-negative integers n, find f(2), f(3), and f(4) for each of…

A: topic - functions

Q: Let p be a prime that does not divide n. Prove that Φpn(x) =Φn(x p)/Φn(x).

A: Given: p be a prime number that does not divide n. To Prove: Φpnx=Φnxp/Φnx.

Q: (4b) Prove that there exists no positive integer x such that 2x < x² < 3x.

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Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: Using Fermat's little theorem, show that if p is prime and (a) if x-1 + yp-1 = zp-1, then p | xyz.…

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Q: (c) Prove that m is prime if and only if y(m) = m - 1.

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Q: If n is an even integer prove that Φ(2n) = 2Φ(n)

A: Given: n is an even integer.To prove: ϕ2n=2ϕn

Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

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Q: X4 none of these X (c) f'(x) does not exist.

A: f''(x) = 0  at sharp corner f'(x) does not exist for f'(x) = 0 , tangent line is horizontal

Q: Prove that for all positive real a we have 22 +5 20x4 2 4/5. 24 x2 +5

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Q: Let n=pp Z+ where p₁ and p2 are distinct prime numbers and 7₁,72 ≥ 1. If o(n) = 80, then determine…

A: Let r1 and r2 be 1.So, n = p1 x p2The totient function, Φ (n) for any value of n is the count of…

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: If P(A) = P(B) = P(AB), show that P(AB + AB) = 0 [AB = An B]. %3D

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Q: Assume that 0  ≤ a1 ≤ 1 and that  an+1=3√(1+an)/2 for all n ≥ 1. Prove that (a)0  ≤ an ≤ 1 for all…

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Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: (a) Prove that for n ≥ 2, 3 > 2n + n². (b) For any n ≥ 1, the factorial function, denoted by n!, is…

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Q: A positive integer n is said to be square-free if it is a finite product of distinct primes. We also…

A: Given that, μn=-1ωn     if n is square free0                 otherwise That is, μ1=1 and for n&gt;1,…

Q: Prove that F2 + F = F2n+2 where Fn is the nth Fibonacci number. %3D n+1

A: Given, Fn2+Fn+12=F2n+2 As we know that, F1=1,F2=1,F3=2,F4=2,F5=4,F6=5

Q: (a) Prove that 1^2+2^2+3^2+………..+n2= n(n+1) (2n+1) divided by 6 For all positive integers n.  (b)…

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Q: A police car is located 40 feet to the side of a straight road. A red car is driving along the road…

A: Please see the attachment

Q: Find functions f and g from the set of positive integers to the set of real numbers such that f (n)…

A: Here, we need to find functions f and g from the set of positive integers to the set of real numbers…

Q: Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such…

A: Given K be any field and a1,a2,...,an be pairwise distinct elements of K. For each i=1,2,3,...,n, pi…

Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: Let n = pq where p and q are two distinct primes. Recall that the Chinese Remainder Theorem states…

A: A good problem. Chinese Remainder Theorem will be used to find F^-1. We present the solution…

Q: p, q, pq | (apq − ap − aq  + a) for all a.

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