10. Prove that if o is the m – cycle (a,a2 ... am), then for all i e {1,2, ...m}, o'(ar) = ar+i, where k +i is replaced by its least positive residue mod m. Deduce that |o| = m.

Q: Show that ø(n²) = nø(n) for every n E Z*.

A: I have used only one formula of euler phi function

Q: Consider the permutation T = (t aq gblvxh)(n s k w fue)(y codjr)(pzmi). (a) Showing your working,…

A:

Q: 4) Show that if |G| = pq for some primes p and q (not necessarily distinct) then either G is abelian…

A: I haved proved using possible cases of order of Z(G)

Q: shor w that To- {0} abelian 7oup. where p an is prime.

A:

Q: Let T(N) = 1. T(floor (N/3)) + 1 and 7(1) = T(2) Prove by induction that T(N) ≤ log N + 1 for all N…

A: Given: TN=TfloorN3+1 T1=T2=1   To Prove: TN≤log3N+1 for all N≥1, using induction.

Q: 1. Prove that if ā and be Zn are invertible modulo n, so is ā - b. Is this the case for ā + b?…

A:

Q: 6. a) Prove that S, is not abelian for all n23. b) Prove that A, is not abelian for all n24. Ay…

A:

Q: Prove that k() = n(1) (n = 1) for integers n and k with 1 ≤ k ≤ n, using a: a) combinatorial proof;

A:

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

A:

Q: 2. Let x (1 3 4) and y= (1 24 3). (Note that these are the same permutations as in th previous…

A: (a) Here given x=1 3 4 =12343241 Thus  x-1=1 4 3=12344213 Because  1234324112344213=12341234=I

Q: - Prove that for all n E Z, we have ged(27n – 20, 4n – 3) = 1.

A: Solution

Q: 4. The generating function f(x) for the Fibonacci numbers 1, 1,2,3,5,8, 13,21, ... is given by OA.…

A:

Q: Show that the last digit of the Fermat number Fn =2^(2^n)  is 7 for all integers n ≥ 2. [Hint: By…

A: We have to prove that the last digit of the Fermat number Fn=22n is 7 for all integers n≥2. For…

Q: 4. Let a 1. 2 Follow these steps to show that (x„)n€N is Cauchy. (a) Draw a picture and let L = b –…

A: Given - Let a < b. Let x1 = a, x2 = b, and xn+2 = xn+1 + xn2 for n ≥ 1. To Show - Use induction…

Q: Let T E Sn be the cycle (1, 2, ..., k) e Sn where k < n. (a) For o E Sn, prove that oTo- (o(1),…

A:

Q: Determine the z-transform of the following: n 20 x(n) = n<0 O · </²/<2 5 3 1-²-(1-22-1) 5 3 1 -z 3…

A: Z transform of the function

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

A:

Q: For every number a e a e {1, 2, 3, ..., 18} which is not a primitive root (modulo 19) find the…

A:

Q: Suppose that phi (x) belongs to Aut (Zn) and a is relatively prime to n.  If phi (a)=b, determine a…

A: According to the given information it is given that:

Q: Use the identity 1/(k(k + 1)) = 1/k – 1/(k + 1) and Ex- ercise 35 to compute E- 1/(k(k + 1)).

A:

Q: 12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show…

A:

Q: b) Suppose n> 1, and m is relatively prime to n. Show that m is a generator of Zn. c) Suppose n> 1,…

A:

Q: 7. Prove by induction. For any n2 0, E F = F,Fn+1• k=0

A:

Q: 3. Prove that 6|(9" - 3") for all n E Z+.

A:

Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: 2n 2n 1/2n + 2 1, 2 n +1 n -

A:

Q: 1. Prove that {2" : n E Z} = {,n :n E Z}.

A:

Q: How would we show that H is cyclic?

A:

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: Consider the elliptic curve group based on the equation where a = 10, b = 8, and p = 11. In this…

A: As per the question we are given the following elliptic curve : y2 = x3 + 10x + 8  mod 11 And we…

Q: 6. Prove that for all n E w, the equality 0 n = 0 holds.

A: We will prove by induction. We need to prove for all n∈w, the equality 0·n=0 holds Let n=1, then,…

Q: Find the Maclaurin expansion of In (cos(x)). О а. x2 x4 x6 17x8 + .. 2520 - 2 12 45 O b. x4 x6 17x8…

A:

Q: 5. Consider the permutations 1234 2 1 3 4 on the set {1, 2, 3, 4). Find fog and go f. (a) (b) F Find…

A:

Q: 1.5 Prove that |e-2z | 0.

A:

Q: In the proof that |(0, 1)| > |N|, we use Cantor's Diagonal Method, where we change the nth digit dnn…

A: We are given the following. In the proof that |(0, 1)| &gt; |N|, we use Cantor's Diagonal Method,…

Q: 15. Prove the following identity C) + (" ; ') + (" * ) + (" ; ) +-+ " : )--- " * ** ') (n + k + k…

A: Binomial Coefficient. The binomial coefficient is the number of ways of picking unordered outcomes…

Q: (5) Show that in a group G of odd order, the equation x² = a has a unique solution for all a e G.

A:

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

A:

Q: . Prove: For n > 1, n − 1 is a generator of Zn.

A:

Q: Obtain the generating function for the number of k-part compositions of n in which the ith part does…

A: Solution :-

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

A:

Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: . Let b, k and m be positive integers that satisfy gcd(b, m) = 1 and gcd(k, ø(m)) = 1, %3D where…

A: Here GCD(b,m)=1 means they are coprime.

Q: 3. Prove that, Σk() = n2n-¹. k=1

A:

Q: 1. Suppose a e Z is an integer. Prove that a is odd if and only if a³ is odd.

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Prove that for all k E N, k ose [coske -sink0j cose Lsinke cosko

A:

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.