Show that (p-1)(p2) (pr) (-1)'r!(mod p), for r= 1, 2, . . . ,p - 1. .

Q: _If A = Verify that [A-!]" = [A"]¬1 ? l1 2.

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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: )"-( 18 6. lim V n4n +(4n)" 2+ n2 4 4+ n2

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Q: 6) Apply induction to show that 2+4+ .2n n(n+1), n 21.

A: We prove this by the induction, firstly prove for n=1, then assume it is true for n=k, and then…

Q: 'rove 1 det(A) = %3D det (41)

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Q: (10) The following element is prime in (Ze,+,.) = .... .. (а) 2 (b) 4 (c) 12 (d) No Choice (а) (b)…

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Q: If al = b (mod p), then prove that a = b (mod p).

A: (ii) given that                                                ap≡bp   (mod p) we have to prove…

Q: 5. (20) Prove by induction that for positive integers n, 4|(5"+1 + 32n-1).

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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: The sum of the first n natural numbers, S = 1 + 2 + 3 + first n natural numbers is 171, determine…

A: Consider the given information:

Q: Let p, q, and r be prime natural numbers with p # q and n= p2q. Then, all the subfields of GF(r")…

A: Answer: Option (C) Given that, p,qand r are prime natural numbers and n is positive integer. Let F…

Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: integer of the form okts is also but not' conversely

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Q: .Prove that la + b + c| < |a] + |b| + ]c]

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Q: п(п + 1) Show that , k 2 k=1

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Q: Give a combinatorial proof for the identity 1+2+3+..+n= (",").

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Q: Prove that Z(D,) # {1} if and only if 2|n (hint: if 2|n look at ri).

A: Let us suppose, Dn=e,r,r2,..,n-1,f,rf,..,rn-1f and nis divided  by 2. Then, n2 is a integer. We see…

Q: Let R be a ring such that for each a eR there exists XE R such that a'x = a. Prove the following :…

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Q: 4. A television producer is looking to release a new series. He believes that the hours per week…

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Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

Q: If P(n) is the statement "n' + n is divisible by 3' Prove that P(3) is true but P(4) is not true.

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Q: If gcd( m, 91) = gcd( n,91) = 1 then show that 91|m³7 – n37 %3D

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Q: 2 (-3) - = (- (-1)² 8 = {) 1 -1 if p= ±1 (mod 8) if p= ±3 (mod 8).

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Q: Prove that if x is real number then L2xd =LxJe bx+!]

A: We will prove the expression by two  steps taking real number x in two form.

Q: Show that 2 (p - 3)! + 1 = 0 (mod p).

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Q: [1+3* 1- 3k] prove that AK = 1- 3* 1+3*] |1– 3* 1+3% | - 5. For A = -1 2

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Q: Let A = {1, 2, ..., 12}. Let aRb mean a = b (mod 5). (a) Give [2]R. [3]R, and [5]R. (b) Give [2]…

A: A set of ordered pairs is referred to as a relation in the area of mathematics and computer science.…

Q: Suppose A = {2, 5, 7, 9, 13, 25, 26}. (a) Find n(A).

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Q: Suppose (1) Compute ||4||₁,||4₂,4„,4F 03 4-(421) 4

A: Given that (1)  The absolute column sums of A are,         First column:     Second column:…

Q: Show that det(A) is (a* – b*)/(a – b), where a +b ab A = 1 a +b ab 1 a +b for a and b any real…

A: Given, A=a+bab01a+bab01a+b