(a) For all integers a, b, and c with a # 0, if a | b, then a | (bc).

Q: If a, b. and c are integers such that a|b and c|b. then ac|b.

A: False

Q: e3x e(n+1)x = 1+ e" + 2 enx Sx 6. n! (n + 1)!

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Q: Prove: For all integers a,b and c, if a|b and a|c, then a|(3b–2c)

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Q: a|b then albc for all integers c.

A: From the question, a|b means that a divides b, and similarly, a|bc means that a divides bc. We need…

Q: 6. Let r> 2 be an integer. Show that R(2r, 2r) > 2

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Q: -. For any integer m and any real number x, if x is not an integer, then [x]+[m- x] = m-1.

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Q: (2) Show that for any integers a,b we always have a? – 4b # 2. -

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Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: If r and s are real numbers and r < s, then there exists a positive integer t such that r+t = s.…

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Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: Let, a1 = 3 and for n > 2, an = 2an-1+ 1, express an in terms of n.

A: a1 = 3 and, an = 2an-1 + 1 for, n≥2 We have to find the generalized term using the recursion…

Q: n Σ k (k!) = (n+ 1)! - 1. %3D k=1

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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: п For all integers n > 1, 1.2 2.3 п(п+ 1) п+ 1

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Q: Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

A: It has been given to us that a, b, c are integers and a2 + b2 = c2.…

Q: (c) Let n be a positive integer and assume that there are integers a and b ab. Show that a < yn or b…

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Q: b) Suppose that A is countably infinite and that there is a one-to-one corresp between A and B.…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: For all integers a, b and c, a|bc if and only if a|b and a|c True or False

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Q: Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 + n², express an in terms of n.

A: In this question, we solve the recurrence relation  in terms of n an=2an-1+an-2+n2witha1=3a2=4

Q: (a) There exists two odd integers whose average is odd. 1 1 (b) There exists distinct positive…

A: (a) ans:- 1,5 (b) ans:- 2,3 &amp;6 For detail...go through the solution

Q: For every pair of integers x and y, if x – y is odd, then x is odd or y is odd.

A: Let x-is even integer and y-is even integer. x=2m, y=2n

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: If a and b are positive integers such that gcd(a,b)=13, then find the sum of all possible distinct…

A:

Q: Let a and b be integers,not both zero .Then a and b are relatively prime if and only if

A: Solution are as follows

Q: For any positive integers m, n (

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Q: For all integers n> 2, 2" < (n+ 1)!.

A: In proof by mathematical induction, we assume that this is true for n=k. Then, we proof that the…

Q: For any integer m and any real number x, if x is not an integer, then [x]+[m- x] = m- 1.

A: Given that m is an integer and x is a real number which is not an integer. we know that for any real…

Q: If x and y are two positive real numbers, then there exists a positive integer n such that ny>x

A: Proof: Suppose there does not exist any positive integer n such that ny&gt;x then ny≤x   ∀n∈ℕ

Q: Give an outline for a proof of "for any integer r and any positive integer d, if d|(x– 1), then d…

A:

Q: Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 +n², express an in terms of n.

A: a1=3, a2=4 n≥3; an=2an-1+an-2+n2 First from the given, the given values shows that the terms are in…

Q: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

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Q: (1) Disprove the statement "For all integers a, b if a | b2 and a < b then a | b." by giving a…

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Q: n fn(x) = D = 1+ x + k! +. . ·+ 2 6 n! k=Q

A:

Q: For any integer m and any real number x, if x is integer, then x]+[m-x] = m- 1. -. not an

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Q: (d))For all integers a, b, and c, if a | b or a c, then a | bc (e) For all integers a, b, and c, if…

A: (d) let for all integers a,b,c if a|b or a|c so a|b or a|c ⇔b = ak1 or  c = ak2  for some integers…

Q: 2. Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 + 5n, express an in terms of n.

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Q: 3. Suppose a is an irrational positive real number. Prove that if x and y are integers and x # y…

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Q: - Let n and r be positive integers such that r≤n. Then (a) P = nx pn-1 (b) P = P-1

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Q: B. Consider the function b(n) = 4n+3 for all positive integers n. Compute b(1), b(2), b(3), b(4),…

A: Sequence: An arrangement of the set of numbers or objects following a rule is called a sequence. We…

Q: Find all positive integers n such that p(n)|n.

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Q: For all integers a, b, and c, a | bc if and only if a | b and a | c. ect one: True False

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Q: For all integers a, b and c, if a divides b and a divides c, then a divides (4b - 3c). True False

A: Answer is TRUE

Q: m above, prove that for every real number c>0, there is a 1/N < c.

A: In every real number c, there is integer N which is N &gt; c.

Q: a) Prove, for all integers k, that 2k-1 is odd. b)prove, for all integers x, if x is odd, then x =…

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Q: a) Find the number of positive integers not exceeding 120 that are divisible by 3 or 4. 1 0 01 0 0 0…

A: As per our guides lines I answered only 1st part

Q: 3. Show that for any real number x, if x2 is odd, then x is odd.

A:

Q: Use the factor theorem to verify the statement

A: Given statement is, x+y is a factor of xn-yn for every positive even integer n.   So, Let fx=xn-yn,…