Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

Q: Show that log, q is irrational for any distinct prime numbers p and q. (Distinct here p# q.) means

A: To Determine :-  logp q is irrational for any distinct prime numbers p and q.

Q: Give a formula for the nth positive even integer: Here is a theorem: 1+ 2+3 + ... + n п(n+1) n = Use…

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Q: n°(n + 1)? 4 j=1

A: According to the given information, it is required to the given equation using mathematical…

Q: For all integers n > 1, E(4i – 3) = 2n² – n i=1 [ In other words: For all integers n > 1, 1+5+9+ ..+…

A: The given series is ∑i=1n4i-3=2n2-n, n≥1 To prove the statement by mathematical induction, we will…

Q: Prove that the sum of (") where i is from 0 to n is 2".

A: According to the binomial theorem, 1+xn=n0x0+n1x1+n2x2+⋯+nnxn……1

Q: Question # 03(c) Prove that neither 2 nor 17 ase Prime elements in Zli].

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Q: If z is an odd ratural number then 2^+1 is divisi ble by 3 hìnt: prove by Induction

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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: 3. Let n E Z be odd. Show that there exist two consecutive integers such that their sum is equal to…

A: Given n be a odd integer. Which is of the form n= 2k+1, k is an integer.

Q: 1: What happens to the above if the root r has a multiplicity m?

A: It is given that, the root r is a root multiplicity m that is fx=x-rmgx where gr≠0

Q: Prove that 2" < n! for any integern 2 4

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: d) If n iš a If t denotes the nth triangular number, prove that in terms of the binomial…

A: The binomial coefficient, nr is defined using the formula, nr=n!(n-r)! r! . The binomial expression,…

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

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Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

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Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

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Q: The small positive integer n for which (1+i)?"=(1-i)?n is =..

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Q: 3 If the characteristic roots of 2 7 are A1 and A, the characteristic root of 7 are - 2 3

A: Given matrix is 3725 with λ1 and λ2 its eigen values.

Q: 6. (a) Find the units digit of 3100 by the use of Fermat's theorem.

A: We need to find the last digit ,therefore find 3100≡xmod 10 According to Fermat's little theorem…

Q: For m an n non-negative integers with 0 < m <n, give a combinatorial proof for the identity EOC) m…

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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: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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Q: Given that P(n) is the equation 1+3+5+7+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we…

A: Principal of mathematical induction

Q: use mathematical induction to prove, for n > 1: 4n3 E(2k – 1)? = | 3 k=1

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Q: 1. Use PMI to show that for any n E N, 4 divides 5" - 1.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

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Q: Prove that the nth Fibonacci number fn≤αn−1, where α is the golden ratio.

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Q: Solve for x. All final answers must be in Zn where n is the modulus. 6x mod 11 ≡1

A: Solving this is equivalent to finding multiplicative inverse of 6 in  Z11 . We have to find x such…

Q: Theorem 9.If k is even and I, ở are odd positivVe integers, then Eq.(1) has prime period two…

A: Given, if k is even and l, σ are odd positive integers.

Q: Show that square root P is irrational where P is prime by using contradiction.

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Q: Prove: If AC [0, 1] and m* (A) > 0, then there exist x and y in Problem 1 A such that x - y) is an…

A: As per our guideline we are supposed to answer only one question. Here 1st question answered.

Q: Is the following statement true or false? For every odd integer n, To answer this question, let n be…

A: Let, n be any odd integer. By definition of odd, n=2k+12, for some integer k So, by solution,…

Q: Prove in a thoroughly written proof that for all integers a  and m, if a and m are the lengths of…

A: Recall the following. Pythagoras theorem: For any right triangle, the square of the hypotenuse is…

Q: Find the Maclaurin serie: e2x

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Q: Show that Σ dn o(n) n for every positive integer n.

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Q: Let pn be nth prime. prove that the sum 1/p1+1/p2+…..+1/pn is never an integer.

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Q: Show that the n-th tetrahedral number O_n is given by S_n= n(2n^2+1)/3

A: A tetrahedral number corresponds to a 3-Dfigure in the shape ofa tetrahedron made of spherical…

Q: 1. Prove by induction that if y = , then d"y = (-1)"n!x¯n-1, dx" where n! = 1 · 2..... n is the…

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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: Use induction to prove or disprove that: (a) Ei-0 to n 2i – 1 = n² Please make sure you show your…

A: Given ∑i=0n2i−1=n2 We have tp rove or disprove ∑i=0n2i−1=n2 by principal of induction method. Let's…

Q: Find the smallest integer i where the statement 10i < 3' is true. Then prove the statement by…

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Q: 1) Guess a formula for the sum 1-3+5-7+..+ (-1)*-(2n – 1) =(-1)-(2i – 1) i=1 and prove your fornmula…

A: The given sum is  1-3+5-7+...+-1n-12n-1=∑i=1n-1i-12i-1

Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

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Q: Solve for z. In 63 = In z + In 7 Enter your answer in the box. z =

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Q: let us assume that /3 is a rational number .show that V3 = where a,b are co – primes b

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Q: Find the solution to 2x = 7(mod 17) that is positive and less than 17.

A: We find solution of given congruence relation

Q: Prove that ei + 01 is odd.

A: Given: e1, e2 are even integers and o1, o2 are odd integers. To prove: e1+o1 is odd integer.