If n is not a poime then there exist Some ab€ 7h Such that ab # Itnk fox all

Q: Not every collection of terms has a GCF other than one. O True O False

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Q: Let a, b and c be integers. (i) If cla and alb then cb. (iv) If cla and cļb then c(sa +tb) for all…

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Q: An integer n is said to be abundant if o(n)>2n. Every multiple of abundant integer is abundant *…

A: We know,  abundant number is a number that is smaller than the sum of its proper divisors. An…

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: a b = a - b is a commutative binanry operation on Z

A: Commutative operation means without changing the operation the values doesn't change.

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

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Q: Let 4; = {0, i} . Find U4, and N 4. i= 1 | A;. i= 1

A: Given :  Ai =  0 , i    To find :  The union ∪i=1∞ Ai    and the intersection  ∩i=1∞ Ai   .

Q: 2. If P (n) : "2.4z + 3""" is divisible by k for all ne N" is true, then the value of k is

A: The solution is given below:

Q: 2. Recall that a number m is said to be cube free if d | m for d > 1 implies that d = 1.…

A: Consider the provided question, A number 'm' is said to be square free it  d2m and d≥1 ⇒d=1 That is…

Q: An integer n > 2 is composite if and only if n does not divide (n - 1)! + 1. :Select one True False

A: The solution is given as

Q: (b) Suppose a, b, and c are integers and suppose a | b and a{c, where a{c denotes that c is not…

A: Given, (i) a divides b  (ii) a doesn't divide c  To show : a doesn't divide (b+c) On the contrary,…

Q: a. Show that if a, b, c, de Zwith a +b√2 = c+d√2, then a = c and b=d. You may use, without proof,…

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Q: Disprove by Contradiction If a,b,c ∈ ℕ, then at least one of a-b, a+c and b-c is even.

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Q: 9. Give a proof by contradiction to show if A and B are sets, then A N (BN A') = {}.

A: Suppose A and B are sets.

Q: Show that /2, /3, 5 cannot be the terms of a single A.P.

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Q: For each statement below decide if it is TRUE or FALSE. • There are two integers n and m such that…

A: We will find out whether the given statements are true or false.

Q: 3. Prove that if a > 1, then a"/n → o.

A: We used binomial inequality

Q: uppose that A ⊆ C so that n(A) = 5 and n(C) = 11. How many subset B, with n(B) = 8 are there such…

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

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Q: O cf AandB are subsets of X such that A=R 3. then fiA = {cB but the converse is not tiug

A: Let A and B be subset of X and  let f be function from A to X If A=B => f(A)=f(B)   taking both…

Q: If A is nonsingular n*n then det(A )= det(A) Select one: O True O False

A: we know the property of the determinant is  det(A-1) = 1det(A) given A is non singular

Q: If A is nonsingular then A' is nonsingular. Select one: O False

A:

Q: 3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d |…

A: To solve the given questions we will use the following formulas : 1.   We say that two positive…

Q: 1. Let, a1 = 7 and for n > 2, an = " express a, in terms of n.

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Q: Prove the statement by induction .+ (4n – 2) = 2n² 2. 1+ 4 + 7 + . + (3n-2) = n ( 3n - 1) 1. 2 + 6 +…

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Q: 1 If n is a positive integer, then (D-a %3D n!

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

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Q: If a²is an even integer, then a is an even integer. If d = ax + by, then d = gcd(a, b). If d|a + b,…

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Q: 2. Prove that n²>2n for n >2

A: Note that, the given statement P(n): n2>2n for n≥2 has little correction because if n=2  then…

Q: t if n is a of by com

A: Given as, n is an integer 3n+2 is even

Q: prove by mathematical induction + E+ 7.16 +.. 1 1 (Зп-2)(3п+1) 3n+1

A:

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: Prove that if k <n and (a^k - b^k) | (a^n - b^n), then k |n

A: We will prove it using contradiction

Q: Prove AX(B−C) = (AXB)−(AXC)

A:

Q: If P(n) is the statement "n' + n is divisible by 3' Prove that P(3) is true but P(4) is not true.

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Q: Let n be a positive integer and M %3D Then M"

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Q: s) Let a and b be positive integers such that b< a < 10b. How many ways are there to place a…

A: In the given problem Balls are labelled and boxes are unlabeled. i.e. we have to distribute a…

Q: 9. Which of the following is true: * N.E. always exist N. E. exist and unique neither necessarily…

A: It is an important part of statistics . It is widely used .

Q: O LUxury Cabin Bear T. NFL Stream NFL Live, H. Question 3 2 pts The difference between any odd…

A: The detailed solution is as follows below:

Q: Cive a paof by induction of the follawig follawing tatementa २६ ० oegee, shea Ifn $.74k+3)=n (2n+s).

A: We need to prove that ∑k=1n(4k+3)=n(2n+5) by Mathematical Induction. The solution is explained…

Q: We say that the mis finite if m (A)s 00. O m(A) = 00, O m (A)< 0. O

A: We can say m is finite only when it is less than than infinite.

Q: If A = {x E N x is odd, x <,7pw many SUBSETS of Acontain no even numbers?

A: If a set contains ‘n’ elements, then the number of subsets of the set is 2n.

Q: 4. Let bo = 1, bị = 3, and for each integer k > 2, we have bk = 4bk–1 – 4bk–2 Prove that for all…

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Q: If A is nonsingular then A' is nonsingular. Select one: O True False

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Q: - Find an open sentence P(n) over the domain S = (3, 5, 7, 9} such that P(n) is true for half of the…

A: Consider the statement P(n): n is divisible by 3. Where n is in the set S={3, 5, 7, 9}

Q: For all n>= 1, then P(n) = n-n is divisible by 3 Select one: O True O False

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Q: if SELECT ALL APPLICABLE CHOICES A) В) Ealn) Ea(n) Ea(n) R-1 is positive and less than a finite one…

A: The objective is to find the correct option.

Q: prove that if a>4 then (a-2)^2>4

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.