A positive integer n is said to be square-free if it is a finite product of distinct primes. We also consider 1 to be square-free For n E N, let ø(n) denote the number of distinct prime divisors of n, and define the arithmetic function µ on N (-1)«(") if n is square-free; µ(n) = otherwise Prove that EH(d) = S1 1 ifn=1 |0 ifn> 1 ulp

Q: 5. Let n Your-Reg-ID (7 digits number) and = let 21, 22, 23,..., Zn be the (distinct) n-th roots of…

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Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: 11.2. (a) If m≥ 3, explain why (m) is always even.

A: phi is the Ruler phi function. We have to prove that phi(m) is always even if m>=3.

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

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

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A: Let x = x0 be fixed given integer. Then,

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A: Let  be a linear transformation from  to  with associated matrix .Let  be a linear transformation…

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: Find all natural numbers in such that n²<z". Prove your assertion.

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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: Let a and b be integers with GCD(a, b) = d and let a = da', b = db'. Prove that GCD(a', b')=1

A: and  be integer such that .It is given that  and .We have to prove .Note: According to the…

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: The matrix for a one-to-one linear transformation from \R^4 to \R^3

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Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: (2n- 2. Give a combinatorial argument to show that k- ways the number of ways to select a committee…

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Q: Prove that for all integers m>0 , m^2+5m+4 is composite.

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

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Q: 5.80 Show that these facts are true. Use the definition of binomial coefficients to show that each…

A: We expand left hand side i.e. nCn nCn= n!/n!*(n-n)! We know that 0!= 1 so  nCn = n!/n! = 1 This…

Q: Prove that If a is an element of order n and p is prime to n, then a\power{p} is also of order n.

A: is prime to We have to show

Q: Consider certain real numbers ui and vi, i = 1,..., n, n <∞0. Show that (u₁v₁ + U₂v₂)² < (u² + u²)…

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Q: prove that (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2n for all integers n greater than or…

A: We have to prove that  (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2 By mathematical…

Q: 5. Identify the mistake(s) in the following "proof": Theorem: The sum of any two positive integers…

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Q: 13. Prove that 6 | (n³-n) for every integer n ≥ 0.

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Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use…

A: As per the question we have the Fermat number Fn = 22ⁿ+1 and its prime divisor p. Here we will use…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.