Let a be a positive integer. Then Lcm(2a+1,2a+2)= *

Q: Abstract Algebra. Please explain everything in detail.

A: To prove (or disprove)  the possibilty of required representations under the given conditions

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Q: Find an expression for An, where n is an arbitrary positive integer.

A: Given: A=1423∈M2×2R

Q: Let a be and integer and b be a nonzero integer. Prove that alb if and only if Mb C Ma.

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Q: Let A = [0, 1] – Q. Determine if A is countable or uncountable. Prove your claim.

A: Let

Q: 6. Let r> 2 be an integer. Show that R(2r, 2r) > 2

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Q: Let a be a positive integer. Then lcm(2a+3,2a+4)= O (2a+3)(a+2) (2a+3)(2a+4) 9a None of these O…

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Q: Let a be a positive integer. Then lcm(2a+1,20+2)= *

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: a|b then albc for all integers с.

A: Statement is ture

Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

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Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: Let a be a positive integer. Then lcm(2a-1,2a)= O None of these O a(2a-1) O a(a+1) O 4a O 2a(2a-1)

A: a is a positive integer.

Q: (c) Let n be a positive integer and assume that there are integers a and b ab. Show that a < yn or b…

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Q: If a, , and c are integers such that albc, then alb or alc. :Select one True False

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Q: Find all integer numbers a for which

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Q: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

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Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

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Q: For an arbitrary integer a, verify the following: (a) 2|a(a + 1), and 3| a(a + 1)(a + 2).

A: We have to verify, 2|a(a+1)  and 3|a(a+1)(a+2) Here we use division algorithm.

Q: Let a, b and m be integers. Prove that if 2a + 3b ≥ 12m + 1, then a ≥ 3m + 1 or b ≥ 3m + 1.

A: Here I have corrected the statement of the Question and provide a detailed solution of the problem.

Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

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Q: a.) Let a and b are positive integers. Prove that (a,b) = 2(a/2, b/2).

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Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

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Q: For an arbitrary integer, say A, verify that 2|A(A+1). Similarly, verify that 3|A(A+1)(A+2).

A: To verify:A(A+1) is divisible by 2.

Q: If a, and c are integers such that albc, then alb or alc. :Select one True False

A: Given integers 'a' 'b' and 'c' such that a|bc then a|b or a|c. This statement is false.  Let us…

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: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

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Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: Let a be a positive integer. Then lcm(3a,3a+1)= O a(a+1) О а(За+1) O 9a O 3a(3a+1) ООО

A:

Q: 8. Let a, b, and c be integers. The show that alb a|bc

A: a, b &amp; c are integers.

Q: Let p = p1 ... Ps be an integer with digits p1,..., Ps. Show that 0.p1 . . · Ps %3D 10° – 1

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Q: Let i be a positive integer and A = [-i + 1 ,1]. Then UA,

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Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: Let i be a positive integer and A, = [-i+1,1[. Then N14= %3D

A: We have to find the first intersection first.

Q: Let n be a positive integer and M %3D Then M"

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Q: prove if a and b are both odd integer, then 16/ (a^4) + (b^4) -2.

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Q: Let a be a positive integer. Then lcm(3a,3a+1)= * None of these За(За+1) a(За+1) 9a a(a+1)

A: Write all factor without repeating term

Q: Vn € Zt [obeb(n, n + 1) = 1]

A: l  let n,n+1 are two consecutive positive integers let  the greatest common divisor i.e…

Q: Let m, n be positive integers. If m n, prove that 5m - 1 5" - 1

A: We show that 5m-1|5n-1 whenever m|n ,for any positive integer m,n

Q: Let i be a positive integer and A, = [-i+1,1[. Then N2, A,- None of these O 10.11 [0,1[ 10.10

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Q: Let a be a positive integer. Then lcm(4a,4a+1) = O a(a+1) O a(4a+1) O 16a O None of these O 4a(4a+1)

A: LCM of two numbers is the smallest number which is multiple of both the numbers. For example, LCM of…

Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.