do I prove this: Let a, b ∈ Z. Give an example of an integer x so that x∣a and x∣b and If x∣a and x∣b, can you give an upper bound on x? i.e. does there exist some m ∈ Z so that x ≤ m?

Q: 2.) Let 1 > -1 be a real number. Prove that (1+1)" >1+ nx for all integers n > 1.

A: We use principal of mathematical induction (PMI) to prove this statement.

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Q: A. Prove that for all positive natural numbers n, 13 + 23 + . + n³ = n²-(n+1)²/4. + ...

A: Explanation of the answer is as follows

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Q: Using proof by induction

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A: Given statement is false because

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A: According to the given information, it is required to show that in any set of n consecutive integers…

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A: As per the guidelines, since there are more than one question only one question will be answered.…

Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

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Q: 1.(a) Do there exist integers a, b, c such that gcd(a, b) = 30, gcd(b, c) = 24, gcd(a, c) = 54?…

A: (a) gcd(a, b) = 30       gcd(b, c) = 24       gcd(a, c) = 54      => 30 | a, 54 | a           24…

Q: How do I prove this? If E is a nonempty set of integers such that inf(E) exists, then inf(E) ∈ E

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Q: Use the principle of ordinary mathematical induction to prove the well-ordering principle for the…

A: To proof the WELL ORDERING PRINCIPLE VIA INDUCTION,WE NEED TO state the principal first as, °The…

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A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

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

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A: prove by contradiction:- to prove it by contradiction, you should start by assuming its negation,…

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Q: #3. Let n be an integer with nz1 and Let b, bz . bn be ... positive real numbers, not all egual,…

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Q: Prove that there is a natural number N such that for all natural numbers n > N, 2/n < 0.1.

A: Assume that there does not exist N such that for any natural number n,  n&gt;N, 2/n&lt;0.1.

Q: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

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.

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Q: 3. Prove by either Well Ordering Principle or induction that for all nonnegative integers,n: + 1) i3…

A: We prove by induction .Case n=1LHS=∑i3i=01=03+13=1RHS=11+122=222=1Induction hypothesis : Assume…

Q: Let n be any integer. Prove that 3n? – 2 is odd if and only if n is odd.

A: Answer

Q: Let n > 1 be an integer. Prove that ℤ/n has exactly n elements.

A: Prove that ℤ/n has exactly n elements. Let n&gt;1 be an integer. That is, n=2, 3, 4, ....∈ℤ ℤ is an…

Q: Prove there are no natural numbers n and p with n > 2 and p² + 1 = 2".

A: ANSWER:

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A: Solution of the problem as follows

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