In the ring of integers, find a positive integer a such thata. <a> = <2> + <3>b. <a> = <6> + <8>c. <a> = <m> + <n>

Q: Use a direct proof to show that if a and b are positive integers, then ab+ba22.

A: Answer

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

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

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Q: Prove that for any integer a, 3 divides

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Q: Find all integers z, positive or negatives, such that z2 + z is prime.

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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: 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…

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: Prove that for any natural number N, there exists N consecutive integers none of which is a power of…

A: Please see the attachment

Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n &gt; 0.

A: Let a be a non zero real number , where m and n are integers…

Q: () is even. 3. Prove that for all positive integers n, the number

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Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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Q: Show that if a, b, c, and d are integers with a and c nonzero, such that a|b and c|d, then ac|bd.

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Q: Prove that for each irrational number r, there exists a unique integer n such that |r – n| < 1/2.

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Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: -ん 2 kodd

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Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r + y-3 is…

A: Solution is in step 2

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: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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

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Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

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

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

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: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

A: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

Q: Prove that for any integer a , one of the integers a,a+2,a+4 is divisible by 3

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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

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Q: Prove the statement: If z E C is constructible, then z has a degree 2 algebraic ex- pression in the…

A: We know that, A complex number z is constructible if and only if there exists a sequence of…

Q: Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A–1 are…

A: Inverse of a matrix A is calculated by,

Q: Prove that if a and b are integers such that a|b, then either a = b or a = -b

A: Take a = 2 ,b = 4 . We have a divides b but neither a = b , nor a = -b.

Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

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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 every integer greater than 27 can be written as 5a + 8b, where a, b e Z+.

A: Let P(n) be the statement, "For n∈ℕ there exist a, b&gt;0 such that              n+27=a·5+b·8 Use…

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

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Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

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Q: For every positive integer a, find a composite number n such that n|a" – a.

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Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

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Q: 2. Show that if a and b are positive integers and alb, then a < b. 3. Show that if a and b are…

A: Basic facts about division will be used in solving this problem.

Q: a2 − b2is divisible by 5.

A: First observe that one's place of the square of a positive number can only one among 0,1,4,9,6, and…

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

A: a&gt;1, then an&gt;1

Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…