Suppose we randomly select a subset of positive integers. How large must the subset be, if we want to make sure that we have chosen at least one pair of integers a, b such that 19|(b − a)?

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

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Q: let a and b be two relatively prime integers. if there exists an m such that (a/b)^m is an integers,…

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Q: The number of permutations of n elements taken rat a time is denoted by nPr and is computed by or…

A: The number of permutations of n elements taken r at a time is given by nPr where nPr is given by,

Q: [12 Let M Find formulas for the entries of M", where n is a positive integer. 88 MT

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Q: For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n +…

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Q: If n + 1 integers are chosen from the set (1, 2, 3,..., 2n). where n is a positive integer, must at…

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Q: . Prove the following statement: For all integers x and y, if x and y are both even, then xy is…

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Q: If a set S has n > 3 elements, how many subsets of S have more than one element, but less than n…

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Q: Let a,b, and c be integers. If gcd(a,b)=1 and both a and b divide c, then prove that a*b divides c.

A: Let a, b, and c be integers. If gcda, b=1 and both a and b divide c, then prove that a×b divides c.

Q: If you have a consecutive integers given by A= {1,2...,k} where k=198 how mnay subsets of A are…

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Q: 8. Let dgcd(135, 351). Find integers m and n such that 135m + 351n = d.

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Q: and at most n coins from a collection f(2n + 1) distinct coins. If the total number of ways in which…

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Q: Find the smallest integer in the given set. Let a, b, and e be integers such that a ≠ 0. Prove that…

A: Let a, b, and c be integers such that a ≠ 0. Prove that if a l bc, then a l c *(a, b). Assuming…

Q: For each divisor k>1 of n, let U,(n) = {x E U(n) | x mod k = 1}. [For example, U,(21) = {1,4, 10,…

A: For each divisor k > 1 of n , Uk(n) =x∈U(n)| xmod k =1. To Find: List the elements of U420,U520,…

Q: Suppose that a divides b and that c divides b. Suppose that gcd(a, c) = 1. Prove that ac divides b.

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Q: Prove that for every n E N is a² | (a + 1)^+1. - 1)n+¹ na — (a + 1), a € N. - -

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Q: Suppose that L and K are subfields of GF(pn). If L has ps elementsand K has pt elements, how many…

A: It is given that, L and K be the subfields of GFpn. If L has ps elements and K has pt elements.…