2. Find integers which are inverse of 17 modulo 24. a) 7 b) 17 c) 13 d) –11 e) –7

Q: Given a set A = {0,1, z} equipped with addition + and multiplication · defined by the tables: + 0 1…

A: We can see that, Here, a×b=b×a. Therefore, Multiplication is commutative. Yes, is the required…

Q: Prove: | a | - | b | <= | a + b |

A:

Q: Show that a permutation o E S, commutes with the transposition (a b) if and only we are in one of…

A: Given: f:S5→S5 is an automorphism. σ∈S5 To do: (d) Prove that σ commutes with the transposition ab…

Q: Find the following using a technique discussed in Section 8.4. 162 mod 45 = 164 mod 45 = 168 mod 45…

A:

Q: By the Femat's theorem, find the inverse of 8 mod 19 A) 817 B) 818 C) 417 D) 418

A: The value of a mod b is the the remainder of a after dividing with b.  The inverse of any integer a…

Q: Prove the statement; a – c is a divisor of ab + cd=→a – c is a divisor of ad + bc.

A: We need to prove the statement; a – c is a divisor of ab + cd => a – c is a divisor of ad + bc.

Q: Given a set A= {0,1, z} equipped with addition + and multiplication - defined by the tables: 千 1 1 1…

A:

Q: Assuming that gcd(a , b) = 1, prove the following: (b) gcd(2a +b, a +2b) = 1 or 3.

A:

Q: Given a set A {0, 1, z} equipped with addition +and multiplication defined by the tables: 0 1 1 1 0.…

A: We have to find

Q: explain these results in APA 7e give as many detail as posible Boxplot-Incongruent Condition 35- 15…

A: From the given output of Boxplot We interpret the outcome.

Q: B) Prove that (O) = ) CE) n –k

A: We use the combination formula to simplify both sides

Q: The elements of order 4 modulo 29 are O 6 and 12 O 7, 14, and 21 O 12 and 17 O None of the choices

A:

Q: Provide a direct proof of the statement below. Let a, b, and c be integers. If a | b and b | c, then…

A:

Q: Prove the following axiomatically. You may assume that + and · on N have the commutative,…

A:

Q: a. Show that if a, b, c, de Zwith a +b√2 = c+d√2, then a = c and b=d. You may use, without proof,…

A:

Q: ind, if possible, AB and BA. (If not possible, enter IMPOSSIBLE in any single o -1 6-[3 12 2] 2 A -2…

A:

Q: . Prove the following: If gcd(a, b) = 1 and ca then gcd(b, c) = 1

A: We will prove the given statement.

Q: 5. Let p = 61. Find all incongruent elements modulo 61, that have order: (а) 5 (b) 10 (с) 60

A:

Q: (S, *) is said to be algebraic binary structure if (A) * is binary operation on S (B) * is…

A:

Q: Let A = (0, 0, #, a) and let # be a binary operation from A X A to A presented by the following…

A:

Q: 2117 mod 77 1317 mod 77 0317 mod 77

A:

Q: (S, *) is said to be algebraic binary structure if (A) * is binary operation on S (B) * is…

A:

Q: 9. Prove the following algebraic identity: (a - b)(a? + ab +b?)=a' -b'

A:

Q: 1 Find the inverse of the matritift eaists, If it does mot erist.wdle DO ES 'NOT EXIST dnd explain…

A: Given: Let A=-27651-2384

Q: Given E denote the set of even integers and O denote the set of odd integers. Let Z denote the set…

A: Given E=Set of even integers and O=set of odd integers And Z is the set of all integers. We have to…

Q: 8. Suppose a is an integer. If 5| 2a, then 5|a.

A: We have to prove that 'Suppose a is an integer. If 52a then 5a.' Suppose that a is an integer and…

Q: Prove by contraposition: if a, b and e are integers and be is not divisible by a, then b is not…

A: Given a,b and c are integers. Statement : P: bc is not divisible by a. Statement: Q: b is not…

Q: Prove the general formula φ(gcd(a, b))φ(ab) = gcd(a, b)φ(a)φ(b).

A:

Q: 8. Give a "train-theoretic" proof of the Fibonacci identity: U11 = u4 · U6 +u5 · U7.

A: Given, u11=u4·u6+u5·u7 Where ui represented the Fibonacci numbers. For train theoretic proof of the…

Q: Proposition 82 (Ex4) Prove: For each integer a, a = 1. if a? = a, then a = 0 or %3D

A: Let a∈ℤ, ℤis the set of all integers. Let a2=a subtract a on both sides ⇒a2-a=a-a ⇒a2-a=0…

Q: Proposition 7. Let a, b, c, d E Z and n be a positive integer. If a = b (mod n) andc = d (mod n),…

A:

Q: 12. The binary operation * is defined on R, the set of real numbers, as a b = a+b+ 1. The identity…

A: For being an identity element , I  => a*I = a ; Now, a*b = a+b+1 (given) ---(I)

Q: Prove that if c is a natural number then for all a and b we have : gcd(ac, bc) = c gcd(a, b)

A: We have to prove that if c is a natural number then for all a and b gcd(ac, bc)=c gcd(a, b) .

Q: Given (N,*) with Va, b e N, a * b = gcd(a, b) a. It is * a law of internal composition b. It is *…

A: Let G be a non-empty set and * be the operation defined on G. Then Law of internal composition: a, b…

Q: Identify the property of integers being illustrated in ( 7)( 5) EI.

A: Here we have to identify the property of integers being illustrated (-7)(-5) ∈ I.

Q: Consider the following statement: "If two numbers have the same remainder modulo 10 then their…

A: If two numbers have the same remainder modulo 10 then their product is divisible by 100. Part(a):…

Q: Let 9={a: aER}, then 9 is Borel algebra.

A: False

Q: Use C.P. and/or I.P. to prove the following. (A ∙ O) ⊃B B ⊃~O A ⊃O…

A: We have to prove using I.P.

Q: Disprove: For all integers a,band c, if a|(bc) then a|b and a|c

A:

Q: Show that a permutation o E S; commutes with the transposition (a b) if and only we are in one of…

A: Hi! As per norms, we will be answering only question (d). If you need an answer to others then…

Q: Show that the axiom A12' implies axiom A12, that is, if A12' is true and {A, B} be a Dedikind's cut…

A:

Q: If AB = -BA, then A and B are said to be anticommutative. Are A = 1 and B = anticommutative?

A: Since, we have been given that A=0-110 and B=100-1,then our aim is to show that whether A and B are…

Q: Let (N,*) where a, b, c E N; a * b = lcm[a, b] a) It is a law of internal composition in N? b) It is…

A: Given :  ℕ, *  be the algebraic structure such that  a*b = lcm[a, b],  where  a, b ∈ℕ. To check :…

Q: Let 9= {[a,b]: a,bER}, then 9 is Borel algebra

A: TRUE ;

Q: 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…

A: Given: A=1101

Q: 12. The binary operation * is defined on R, the set of real numbers, as a b = a+b+ 1. The identity…

A: For being an Identity ; k a*k= a  --(I) So, a*k = a+k+1 ;     (given) From (I) a = a+k+1 => K =-1…

Q: * is said to be commutative in A for all a,b ɛ A Select one: O a a*b b * a O b.a * b = b * a Oc a+b…

A: * is said to be commutative in A for all a, b in A, if, Option (b) a*b = b*a

Q: Let a, n ∈ N. Show that if an − 1 is prime, then a = 2 or a − 1 is prime and n = 1 or n is prime.…

A:

Q: 42 Consider the following diagram. netric definition for 30 AD

A: