Prove that if a and b are integers of the form 4n+1, then ab is of the form 4n+1.

Q: If a is an algebraic integer satisfying a³ + a + 1 = 0 and ß is an algebraic integer satisfying B² +…

A: Note: There are many ways of proving. We have done by giving the proof more elaborately for the…

Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: 10) Let a and b be positive integers. Prove that (a, b) = a if and only if a divides b.

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Q: Prove that there exists an odd integer m such that every odd integer n with n greater than or equal…

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Q: Prove: there exist no integer a and b for which 21a + 30b = 1.

A: We can prove this by contradiction. Take the negation of the statement and check if it is true, if…

Q: Prove: For all integers a,b and c, if a|b and a|c, then a|(3b–2c)

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Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

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Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

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Q: If all entries of A are integers, and det A = 1 or -1, prove that all entries of A -1 are integers.…

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Q: Prove that there are no positive integers a and b such that a + b³ = 250.

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Q: Prove that if a > 2 and b are integers, then a † b or a { (b+1).

A: Given that a≥2 and b are integers. We will prove it by contradiction. Let a|b and a|(b+1) then there…

Q: (2) Prove that there are no integersm and n such that 7m + 2ln - 15.

A: There are no integers m and n such that 7m+21n=15

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: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: Let a, b eN. Prove that if a + b is even, then there exists nonnegative integers x and y such hat x²…

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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: Prove that there do not exist integers m and n such that 12m + 15n = 1.

A: Proof by contradiction. Assume the given statement is true.

Q: Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

A: It has been given to us that a, b, c are integers and a2 + b2 = c2.…

Q: Prove there exist no integers a and b for which 18a + 6b = 1

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Q: Prove that for all the integers a and n, if a | n, then a^2 | n^2

A: Given that a | n. This means n=ak, where k=integer. Then we square both sides: n2=(ak)2 or n2=a2k2

Q: Prove that for any integers a and b, if a b and alc, then a|(3b-5c).

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Q: Show that if a and b are coprime integers, then gcd(a + b, a² – ab + b²) = 1 or 3.

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Q: Prove that there exists two integers a and b such that a+b > ab

A: We want to prove that there exists two integers a and b such that a+b>ab

Q: Prove that for all positive integers n, 4n-1 is divisible by 3.

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Q: 3. Show that if a and b are integers such that alb, then ak|b* for every positive integer k.

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Q: Use the contradiction method to prove: There exist no integers a and b for which 21a + 30b = 1

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Q: Show that is a and b are positive integers such that (a | b) and (b | a), then a = b

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Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

Q: Prove that there are no integers m and n such that m2=4n+2.

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Q: Prove that the statement is true for every positive integer n.

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

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Q: Prove that if a and b are both odd integers, then 8 (2a2+36²).

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Q: Prove that if the product of two integers qs is odd, then q2 + s2 is even. Then show that for…

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Q: Prove that if ∣A∣ = 1 and all entries of A are integers, then all entries of ∣A−1∣ must also be…

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Q: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

A: Given that, a and b be integers and in both of them at least one is non-zero. And, greatest common…

Q: Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 +…

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Q: Prove: For all integers a, b, and c, if alb and alc then a (2b+3c)

A: prove is given below ....

Q: Prove that there do not exist positive integers a and n such that a? + 3 = 3".

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Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

Q: and

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Q: Prove that for any positive integer n, Vn is either an integer or irrational.

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Q: Prove by contradiction that 6n + 5 is odd for all integers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Prove that there exist integers m and n so that 2m+7n=1.

A: The  integer is a whole mumber that may be positive number or negative number or zero.But not a…

Q: Prove or disprove:  For all integers a and b, if a+b is even, then a is even and b is even.

A: We disprove this statement by taking a counter example.

Statements and Logic

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