2. Let m, n, a, b € Z. Prove that if 6 | m and 6 | n then 3 | (am + bn +15).

Q: if a | b and a | c, then a | (mb + nc). | C,

A: If a|b then b=ka for some integer k If a|c then c=ga for some integer g

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

A: .

Q: Let U = {1, 2, 3, 4, 3, G, 7, 0, 9, to ; A = { positive even Factors OF to; {1,3,8,9} and {x Ix ic a…

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Q: Let A= {1,2,x,y,z}, and B= {1,2,3,z}. How many distinct elements are in (A*B) intersect (B*A

A: Given that A=1,2,x,y,zB=1,2,3,z To find the number of distinct elements in A×B∩B×A:

Q: 4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b)…

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Q: Prove that (ab + cd)² ≤ (a² + c²)(b² + ď²) for all a, b, c, d ɛ R.

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Q: 4. Let A = of O 2| Calculate

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Q: (а b)с + (а - с)b + (с b)a 3 ve that v belongs to B.

A:

Q: 5. If x is an integer, then x' 2x. b d a C 6. There exists a real number x such that x'= x . FL

A: we can solve this problem using some basic properties of integers and real number.

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

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Q: Let a e Z. Prove that if 6 | a and 10 | a, then 15 | a.

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Q: 12. Prove that if d divides a and d divides b, then d divides ax + by for all integers a and y.

A:

Q: Reject Ho if |z| > N

A: We know that  Null hypothesis H0. Alternative hypothesis H1.

Q: Let a, b € N. Choose all correct statements. O gcd(-a, b) = gcd(a, -b) O ged(2a – b, a + b) =…

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Q: Let a, b, n E Z. Prove: If a|n and b|n with gcd(a, b) = 1, then ab|n. (I

A:

Q: Prove ImA = A = Aln if A is m X n

A: Given: A be m*n matrix

Q: · (b + c) = ² 7. Prove that à. = a · b + ac for all a, b, c € R³.

A:

Q: 1. Let a, b € Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 46.

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Q: E. Show that both R and R² may be covered by countably many open balls.

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Q: 11. Prove that if d divides a, then dk divides ak for every positive integer k.

A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

Q: Prove that |A U B| + |A n B| = |A| + |B|

A: Let A and B are two sets. A = number of elements in the set A. B = number of elements in the set B.…

Q: Let A = { {a}, {b}, {c}, {d}, {e} } Determine by extension P(A

A:

Q: Suppose 50 percent of the customers at Pizza Palooza order a square pizza, 72 percent order a soft…

A: Given information- Percentage of the customers order a square pizza = 0.50 Percentage of customers…

Q: Lise a direct proof to show "fx's odd and y is even, anen x+1s odd."

A: The given statement is: "If x is odd and y is even, then x+y is odd" Result: Any odd number x can be…

Q: 9. Let a, b, and d be positive integers. Prove that if (a, b) = 1 and d divides a, then (d, b) = 1.

A:

Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: 10.

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Q: Suppose a, b, c, d e Z, a + b = c, and dla. Prove that d|b if and only if d|c. 5.

A: see below the calculation steps

Q: why do we assume a|b and prove that way instead of ac|bc and prove back to a|b

A: why do we assume a|b and prove that way instead of ac|bc and prove back to a|b .

Q: Suppose gcd(ab,c)=1. Prove that abc | d if, and only if ab | d and c | d.

A:

Q: Let A; = [i, i + 1], i = 1,2, 3, 4,... Then A n A2 N A3n A4 is O A. {1} O B. { } O C. [1,5] O D.…

A: We have Ai = i, i+1 and i = 1, 2, 3, 4 (we need to consider only up to 4 since we have up to A4…

Q: . Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A2 + B2). (a) Prove the statement.…

A: 5. Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A^2 + B^2). (a) Prove the…

Q: Let a e Z. Prove that if 7 | 2a, then 7 | a.

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Q: - If A = {2,3} and B = {x | x ≤ Z+,x <3}, then A x B= Select one: O a. {(1,2), (1,3), (2, 2), (2,3)}…

A: Given that the setsA=2,3 and B=x | x∈Z+ , x&lt;3

Q: 3. Let a, b, c € Z. Prove that if a³|b and b5 | c, then a¹5| c. 4 Prove that for any n = 7 3m² + 112…

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Q: Let Ai = = [i, i + 1], i = 1, 2, 3, Then A₁ A2 A3 A4 is. a. {1,2,3,4,5} b. (1,5) d. [1,5] ....

A: We have given that Ai=[i,i+1]A1=[1,2]A2=[2,3]A3=[3,4]A4=[4,5]