Using the definition of the "divides" relation discussed in class, determine which of the following statements are true. Select all that apply. -4 | 16 5 | 20 3 | 0 15 | 3 1 | 8 7 | 22 0 | 6 4 | -16
Added by Virginia B.
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The notation "a | b" means that "a divides b" if there exists an integer k such that b = a * k. Let's evaluate each statement step by step. ** Show more…
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3. Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For all integers a, b, and c with a ≠0, if a | b, then a | (bc). (b) For all integers a and b with a ≠0, if 6 | (ab), then 6 | a or 6 | b. (c) For all integers a, b, and c with a ≠0, if a divides (b - 1) and a divides (c - 1), then a divides (bc - 1). (d) For each integer n, if 7 divides (n^2 - 4), then 7 divides (n - 2). (e) For every integer n, 4n^2 + 7n + 6 is an odd integer. (f) For every odd integer n, 4n^2 + 7n + 6 is an odd integer.
Keondre P.
Select the following statement that is TRUE: A number k divides the sum of three consecutive integers n, n + 1, and n + 2 if and only if it divides the middle integer n + 1. An integer n is divisible by 6 if and only if it is divisible by 3. For all integers a, b, and c: a|bc if and only if a|b and a|c. For all integers a, b, and c: a|(b + c) if and only if a|b and a|c. For all integers a and b: a|b if and only if a^2|b^2.
Adi S.
Does 17 divide each of these numbers (just answer yes or no)? 68 b) 84 357 d) 1001 Show that if a | b and b | a, where a and b are integers, then a = b or a = -b. Show that if a, b, c, and d are integers, where a != 0, such that a | c and b | d, then ab | cd. Prove that if a and b are nonzero integers, a | b, and a + b is odd, then a is odd. Prove that if a is an integer that is not divisible by 3, then (a+1)(a+2) is divisible by 3.
Sri K.
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