1 Consider the element A = in SL(2, R). What is the order of 1 1 A? If we view A = as a member of SL(2, Z,) (p is a prime), 0 1 what is the order of A?

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

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Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: 5. Let a = 2, az = 4, and an+2 a, = 2" for all natural number n. = 5a+1 - 6a, for all n 2 1. Use the…

A: Given information: The given values are a1=2,a2=4 and  an+2=5an+1-6an for all n≥1. To prove : an=2n…

Q: Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n > 3, an = 2an-1+ bn-1 for n > 2 bn = bn-1+ ɑn-1 express…

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Q: Prove that for a # -1, a>0,n EN IS (1+a)^ n >=1+na.

A: Here let A=(1+a)n, a≠-1, a>0, n∈N and B=1+na

Q: Make an outline for a proof by induction that 2 = n(n+1)(2n + 1) for all n E Z+. k=1

A: Note: We can solve one question at a time so kindly upload second question as another question.

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: Let p be a prime. Show that there is exactly two elements a belongs to Zp s.t a^2 = 1.

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Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

Q: 1. Vn EN, Prove the following by mathematical induction. M= 1 =2 - n+2 2n

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Q: R² = gen gen

A: see below the reason

Q: 5. Prove the following identity for n 2 1: Fzn+1 = F+1+ F;

A: We need to prove that ,  F2n + 1 = F2n+1 + F2n We know that ,   Fm+n = FmFn+1 + Fm-1Fn ,  m ≥ 1 , n…

Q: 5. Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n ≥ 3, an = 2an-1+ bn-1 for n ≥ 2 bn = bn-1 + an-1…

A: Given Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n ≥ 3, an = 2an-1+ bn-1 for n ≥ 2 bn = bn-1 + an-1…

Q: Prove by induction that (2n+1)3 is an odd number for all positiveintegers n

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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: 4. Let n E N. Find all elements a Zzn+1) of order 2.

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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: 4. Let bo = 1, b1 = 3, and for each integer k > 2, we have br = 4bk-1 – 4bg–2 %3D Prove that for all…

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Q: Let n E Z', where n > I. Show that if n is a product of distinct Mersenne primes, then o(n) = 2k for…

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Q: 1. Suppose that an < bn for all n, an → L, and bn → M. Show that L < M.

A: As per our guideline we are supposed to answer only one question.kindly repost other question. The…

Q: 11. Given: LM NO; ZLMO = ZNOM Prove: LO NM N. Statements Reasons

A: Solve the following

Q: السؤال 2 For all n E N, n> 1, there exists a prime p such that p|n. ylan O İhi O

A: True

Q: 4. In Z[V2] = {a + by2ļa, b e Z}, show that every element of the form (3+ 2v2)" is a unit, where n…

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Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b € N, and odd c€ N such…

A: Since n is even and non-zero . Since n is non zero ⇒ n is either positive or negative. Let's write…

Q: Prove the following in Z[i]: if a y and 3|y with a and B relatively prime then aß | y. in

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Q: Find the elements of Z24 of order 8. (Hint: o(1)=24)

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Q: 4. Let bo = 1, b = 3, and for each integer k > 2, we have br = 4bx–1 – 4bk–2 Prove that for all…

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Q: Give a combinatorial proof: Forn and k satisfying 1 <k < n, n nk = Ek · (j – 1)k-1. j=k

A: The complete solutions are hoben below

Q: Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n ≥ 3, an = 2an-1+ bn-1 for n ≥ 2 bn = bn-1 + an-1…

A: The Fibonacci sequence can be defined as a series where a term of the sequence is the sum of the…

Q: In the domain of Gaussian integers Z[i], the element 15 is 青 O Reducible O Unit O Irreducible O None…

A: Since 3∈ Z[i] and 5∈ Z[i] And 15 = 3 × 5 Hence 15 is reducible element in Z[i]

Q: 2. Let m e N, then show that (a) For integers x, y, x = y(mod m) if and only if the remainder when x…

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Q: SThe foiloLoi with N=10: 75 11 8362198 aCnlcone the popo n han men ng 15nsct Of chh for n popolahon

A: Given data set is7,5,11,8,3,6,2,1,9,8Let "x" denotes the give data setx={7,5,11,8,3,6,2,1,9,8}

Q: If nis a positive integer 1 then prove (D - a) n!

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Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: The expansion of esin(x) is Select one: O a. x² x4 1+x+ ... 8 Ob. x² 1+x+- 2 - t... 8 c. x5 1+x+ 6.…

A: To find the expansion of

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZ≈ Z/mnZ if and only if m and n are…

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Q: .In Z[V-5], the element 29 is: O Unit O reducible Irreducible None of the choices

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Q: 3. (i) Give a definition of Fermat's Little Theorem. (ii) For all n E Z, show that 30 | (nº – n).

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Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: 9. Let n E Z. Prove that 3k EZ such that n' = 8k or n' = 8k + 1. In other words, prove that by…

A: We will use the fact that any integer is either even or odd to prove the required result.

Q: Let n = pq where p and q are two distinct primes and consider Zn. Further, let [a] E Zn be an…

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Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: 3. Prove that if p is a prime >3 then the numerator of 1 + 1/2+1/3+...+V/(p-1) is divisible by…

A: Given: Let us consider the given, r=1+12+13+...+1p-1

Q: 2) use the mathematical induction to prove : A) 4" - 1 divided by 3 , Vn E N+

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Q: 6.31 Boos and Hughes-Oliver (1998) detail a number of instances where application of Basu's Theorem…

A: (a) Let X1, X2.......Xn be the independently identically distributed randm variables follows nμ, σ2…

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

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Q: . In Z[V-5], the element 29 is: None of the choices Irreducible O Unit O reducible

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Q: 3. For every integer k > 0, define A = {2*n : neN and n is odd}. Prove that {Ag : k = 0, 1, 2, 3,…

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Q: 14. Let a/b be a fraction in lowest terms with 0 <a/b<1. (a) Prove that there exists n e N such that…

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