Prove or disprove: There exists an integer a such that 14 | (3a — 5) and 21 | (2a + 3).

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: Prove or give a counterexample to the following statement. For each positive integer a, there exists…

A: Given: 12b2+b<1ab2To check whether the given inequality is true or false keeping in mind for each…

Q: Prove that, nonetheless, for all integers a, if 2 | a and 3 | a, then 6 | a

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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 5 does not divide 5n+3 for any integer n.

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Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

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: Consider the two statements: P: For every integer n there exists an integer m such that m + n = 0.…

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Q: Prove that the product of two integers, one of the form 3k₁ +2 and the other of the form 3k2 +2,…

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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: Knowing that for every real numbers a and b, (a + b)² = a² + 2ab + b², one can easily prove that for…

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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: Prove that there are integers x and y such that 10x-13y=1

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Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

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

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Q: Prove for every integer n > 8 that there exist nonnegative integers a and b such that n = 3a + 5b.

A: Proof by induction For the base case: n=8, we have 8=5+3. Suppose that the statement holds for k…

Q: Select the following statements that are true. gcd (2022, 111) = 1 2023 is a prime gcd (2022, 1685)…

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Q: Let a and b two positive integers. Then we have that axb LCM(a,b) = GCD(a,b). O True O False

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Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

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Q: prove that for every even integer n, 2n-1 is odd

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Q: Prove that n is odd if and only if 3n+9 is even. You must use definitions of even (and odd, if…

A: An even number is an integer that is divisible by 2, i.e., when divided by 2, the remainder is 0.An…