(a) Show that if m and n are integers such that gcd(m, 4) = 2 and gcd(n, 4) = 2, then gcd(m +n, 4) = 4. %3D %3D

Q: Suppose gcd(a, n) = 1. Show that there exists b such that [a]n * [b]n = [1]n.

A: Using division algorithm.

Q: gcd(a, b) = 1 → a and b are relatively prime. %3D

A: Relatively prime numbers: Any two integers a and b are said to be relatively prime if and only if…

Q: 5.2.8 suppose that a store offers gift certificates and denominations of $25 and $40. Determine the…

A: The given denominations are $25 and $40.Here, the given denominations are multiples of 5.The amounts…

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: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: 5 Find the coefficient of x in the expansion of x- The coefficient of x is

A: The general term of expansion of (a+b)n  Tr+1=Crnan-rbr Here, a=x, b=-2x, n=14 So, Tr+1=Cr…

Q: (a) Let m and n be integers greater than 1. What is the order of the element (1,1) in Z/mZ x Z/nZ?…

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Q: 3. Let A={n €Z, 2Z). Prove or disprove: For every n E A we have n² = 4.

A:

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: a) What is the generating function for {ak}, where akis the number of solutions of x1 + x2 + x3 = k…

A:

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: 3. Give combinatorial proof of a k (") = n () k-1

A: We know that the combination formula: C(n,r) = n!/(n-r)!r!

Q: 2- (a) Equating the coefficients of xk+r-1 to zero Σα. +r)(k +r - 1)ax xk+r-2 - [(k +r)(k +r- 1) +…

A:

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: 3. Let n e N. Prove by mathematical induction that > 3* = 4". k k=0

A:

Q: Give a combinatorial proof of the identity: -1 k(²) = n(n=¹) k-1.

A: Use the following combinatorial formulae, to obtain the result. nr=n!r!n-r!n!=nn-1!

Q: 1- Let n and r be positive integers such that r≤n. Then (a) P = nx P-1 (c) P = pn a O b O d n-1 (b)…

A:

Q: 4. Let, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an-1 + an-2 + n2 , express an in terms of n.

A: As per guidelines we should able to do only one question at a time because both questions are…

Q: Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n Σ'=1+2+...+ B₁=k²= 1² +2²+. k=1…

A:

Q: Solve the recurrence relation for computing the binomial coefficients (below): c(n, k) = c(n - 1, k…

A: The formula of combination : Crn =n!r!×(n-r)!           for n≥r To show Ckn = Ck-1n-1 + Ckn-1…

Q: (a) Find all the irreducible representations of S3.

A: To Determine :-  All the irreducible representations of S3.

Q: 1- Let n and r be positive integers such that r≤n. Then (a) Pn=nx pn-1 (c) P = pn n-1 (b) P = P-1…

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

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Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: #2. The binomial theorem states that for any real numbers a and b, In (a + b)n = =Σ" ("a²-kbk for…

A:

Q: a 1 In parts (a) - (c), prove that for all n E N: (a) n-1 k k=0 (b) ¿(:)-- 2n+1 k k-0 (c) E(-1)* k=0…

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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: Write all elements of S4. Show that S4 has no elements of order > 5.

A: The given question is related with abstract algebra. We have to write all the elements of S4 . Also,…

Q: 5. Find the generating function to determine the number of ways to pick k objects from 2n objects…

A: Given: When duplicates are allowed and the (2i – 1)th and (2i)th objects occur the same number of…

Q: 25. Prove the identity 2 (2n k k=0 by counting the number of subsets of size n of the set {a1, a2,…

A: We need to show that ∑k=0nnk2=2nn by counting the number of subsets of size n of the set a1, a2, … ,…

Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

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

A: Consider the given condition,  n∈ℤis a non-zero and even. The objective is to prove that there exist…

Q: (5a) Let A = {3, 4} be a subset of S = {1,2,... ,6} and let n E S. || n2(n+1)² Prove that if is…

A:

Q: 7. Construct the generating functions for the number of partitions of n such that: a. No part is a…

A: Determine the generating functions for each part. Write the generating function of part (b) and then…

Q: 1. Use mathematical induction to show that n2 – 3n +4 is even for all n > 1.

A: To show that n2-3n+4 is even for all n≥1, using Mathematical induction.

Q: 3. Let h be a prime number and let k be some integer. What are the possible values of gcd(h + k, h -…

A:

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: a) What is the generating function for {ak}, where akis the number of solutions of x1 + x2 + x3 + x4…

A:

Q: Fix n e N. Prove that gcd(5n + 3, 3n + 1) = 1

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 by induction:  gn=gn-1+gn-2+gn-4   for n greater than or equal to 4, with g0=1, g1=1, g2=2,…

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

A: Solution :-

Q: 2. (a) Let n E N. Determine the set of all integers which leave a remainder n+1, n+2, n+3 when…

A: 2. (a) Consider the given information. As, We have given that,  2n+1,2n+3 and 2n+5 are consecutive…

Q: 7. (a + b)"= E=o( n choose k) a™-kb* for any integer n>=0. Use this theorem to compute (2x-3)* ) The…

A: Given, 2x-34

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: 3. If the coefficients of 4 consecutive terms in the expansion of (1+ x)" are k1, k2, k3 and k4…

A:

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

A:

Q: 1) By using the principle of mathematical induction, prove that 1 + 3.5 n Vn E N 1.3 (2n-1)(2n+1)…

A:

Q: 4. Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2+n², express an in terms of n. 5. Let, a1 = 1,…

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.