2. Given integers a, b, c, d, verify the follOWilng. (a) If a | b, then a | bc. (b) If a | b and a | c, then a² | bc. (c) a |b if and only if ac | bc, where c 0. (d) If a lb and cld, then ac | bd.

Q: Prove the following statements: iii. Suppose that a, b, c and d are positive integers, prove that…

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Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

Q: 1. Give a proof of the following statement. 3 integer c > 0 V integers r > 0 3 integer d > 0 : x² –…

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

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Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: Let a, b, c be positive integers and let d = (a, b). Since d|a, there is an integer h with a = dh.…

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Q: Provide a direct proof of the statement below. Let a, b, and c be integers. If a | b and b | c, then…

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Q: (a-b)’ (b-c)' P (-9) a 3 d

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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: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Prove the following statement: "Let a, b and n be integers. If n doesn't divide ab, then n doesn't…

A: The given statement is -" Let a.b and n are integers. If n doesn't divide ab, then n doesn't divide…

Q: For all intergers a,if 6|5a,then6|a. Prove or disprove this statement."|"means divides

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Q: Show via proof that for two positive integers a, b where a 2 b, (use combinations e.g. n choose k)…

A: We shall prove for the given statement. Though it has little ambiguities.

Q: 18. Prove the following proposition: For all integers a and b, if 3 divides (a? + b?), then 3…

A: Suppose 3 divides a2+b2 for all integers a and b To prove 3 divides a and 3 divides b Let us prove…

Q: 4. Let A C BCR. Prove A° C B°.

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Q: Prove: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

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Q: If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that

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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: Let a, b and m be integers. Prove that if 2a + 3b ≥ 12m + 1, then a ≥ 3m + 1 or b ≥ 3m + 1.

A: Here I have corrected the statement of the Question and provide a detailed solution of the problem.

Q: Determine whether the following statements are True or False: A) For all positive integers a,b,…

A: using :- if a|b then a=bq , where q is unique integer.

Q: Prove the following. 2. If alc and a+ b = c, then alb.

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Q: 3. Let r and y be positive integers. 4. Let a, b, c e Z. If c divides a +b,

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Q: Prove by Contradiction: There are no integers x and y such that = 4y+2.

A: We have to prove the statement "There are no integers x and y such that x2=4y+2" by contradiction.…

Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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Q: Suppose that a, b, and c are integers. Prove that if a2 +b2 =c2 then a or b is even.

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Q: Prove that ord,(a + b) > min(ord,a, ord,b) with equality holds if ord,a ± ord,b.

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Q: let a, b, and m be integer, prove that if 2a+3b>=12m+1, than a>=3m+1, or b>=2m+1

A: To prove: For integer a, b, and m if   2a+3b≥12m+1Than, either a≥3m+1, or b≥2m+1.

Q: Prove or disprove: For any integers a,b,c, if a { b and b X c, then a X

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Q: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

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Q: 3. Suppose a, b, c E Z. Prove that if a? + b = c²,, then a is even or b is even.

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Q: Show that if h+1 distinct integers are, Chasen from the Set El,2,en.,4n}. then there are always two…

A: To show that choosing n+1 distinct integers from the set {1, 2, 3, ..., 4n} then there are always…

Q: 2. Use a proof by contrapositive to show the following: Suppose a, b e Z are integers. If a² (b² –…

A: To Proof- Use a proof by contrapositive to show the following: Suppose a, b ∈ ℤ are integers. If…

Q: Define sets A and B as: : {n E Z\n = 4r + 3 for some integer r}. • B= {m E Z m = 4s – 1 for some…

A: the objective is to solve the above questions.

Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

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Q: Let а, b and c be integers. Use the method of proof by contradiction to prove that if a|b and a f c,…

A: We have to solve given problem:

Q: Disprove: For all integers a,band c, if a|(bc) then a|b and a|c

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

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Q: Show that if a,b, and c≠0 are integers, then a∣b if and only if ac∣bc.

A: We have to prove the following statement -  If a,b, and c≠0 are integers, then a∣b if and only if…

Q: (b) If r23= 1, prove that R² (23) = 112=113 and 01.23 = 01 (1-12).

A:

Q: 7. Establish for A is m × n and B, C are n x p, that A(B+C) = AB + AC

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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: for all integers a, b, and c, if a divides b and a divides c, then a divides (b2-c2). prove this…

A: Introduction: Divisibility is the base of number theory. For two integers a and b, if there exists…

Q: 4.2. Let a, b € Z, where a + 0 and b + 0. Prove that if alb and bla, then a = b or a = -b.

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Q: Let A = {a,b, c} and B = {b, c, a}. Prove that A = B. Thus, the order %3| in which the elements are…

A: The given sets are A={a,b,c} and B={b,c,a}we have to show that A=BResult : two sets A and B are…

Q: tet r and m be integer, prove that 17/20+3m iff, 17/9n

A: Solution

Q: Prove that if p,q, and r are integers, r|p and r|q, then r^2|pq.

A:

Q: 1. Show that the set of integers S = {a² + b2 : a, b € Z} that can be respresented as the sum of two…

A: See the details below.

Q: (a) For integers a, b, q, r, prove that if a = bq + r then (a, b) = (b, r).

A: Since you have posted a multiple question according to guildlines I will solve first(a)    question…

Q: Show that det(A) is (a* – b*)/(a – b), where a +b ab A = 1 a +b ab 1 a +b for a and b any real…

A: Given, A=a+bab01a+bab01a+b

Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.