If a, b e Z such that a b then the numbers a and b are called relatively prime numbers if gcd(a, b) = 1

Q: 2. Let N = {A, | ne N}, where A, = {n,n+ 1, n +2, ...}. (a) Find UN and prove your conjecture. (b)…

A: we have given Ω = An n∈N where An =n, n+1, n+2,...... we have to find (a) find ∪Ω and prove your…

Q: Suppose gcd(a, n) = 1. Show that there exists b such that [a]n * [b]n = [1]n.

A: Using division algorithm.

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

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Q: Suppose p is prime and that a,b EN with a, b > 1. (a) Show that if p|ak for some k E N, then p|a.…

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Q: gcd(a, b) = 1 → a and b are relatively prime. %3D

A: Relatively prime numbers: Any two integers a and b are said to be relatively prime if and only if…

Q: Show that if a\b and gcd(b, c) = 1 Then gcd(a,c) = 1 %3D

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Q: (a) Prove that 3n2 +1 is even if and only if n is an odd integer. (b) Suppose A, B,C and D are sets.…

A: The given question has two parts. The solution is explained below.

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 if gcd(a, b) = 1, then gcd(a + b, ab) = 1.

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Q: If a, bEZ such that a b then the numbers a and b are called relatively prime numbers if ged (a, b)…

A: True

Q: Assuming that gcd(a , b) = 1, prove the following: (b) gcd(2a +b, a +2b) = 1 or 3.

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Q: Let a, b, x E Z be given. Show that: ax + 6 even (x odd Vb ev en )

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Q: Prove that for a # -1, a>0,n EN IS (1+a)^ n >=1+na.

A: Here let A=(1+a)n, a≠-1, a>0, n∈N and B=1+na

Q: 2. Let a € R and n € Z0. Show that of the numbers a, 2a, distance from an integer is at most 1/n.…

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

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

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

Q: |Let a, b, and c be integers. Prove that if gcd(a, b) | bc, then a | 1 and gcd(a, c) 1 and a (a) C.…

A: To prove the required divisibility properties under the given conditions on the greatest common…

Q: Given positive integers k and m, there exists an integer N ≥ m such that N →(m)k

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Q: Find a natural umber N such that if |a – 5| < e <1 then |a² – 25| < Nɛ.

A: According to our company policy, we are supposed to answer one question at a time. For the rest…

Q: 8 ) ) Let A=[0 ;], B=[; %], C-[; !] C=[o a) A &B& C are nilpotent in M₂(R) b) A &B are nilpotent in…

A: Nilpotent element: An element x of a ring R is called a nilpotent element if there exists smallest…

Q: Prove that ∀a, b, k ∈ Z that ((4|k) ∧ (k = ab)) ⇒ (2|a ∨ 2|b)

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Q: (1) Prove that is a, b, and c are integers with c + 0 then ac = bc = a = b ( that is this a key step…

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

Q: If a and b are even then GCD(a,b) must be 2

A: Given statement :- if a and b are even then GCD(a,b) must be 2

Q: If a and b are integers with gcd(a, b) = d Then gcd d' = -1 d/

A: Given a statement

Q: 4. Let bo = 1, b1 = 3, and for cach integer k> 2, we have bz = 4bg–1 - 4bg-2 Prove that for all…

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

A: According to our guidelines, we can solve only the first question. Please repost other questions as…

Q: 4. Let bo = 1, b1 = 3, and for each integer k > 2, we have br = 4bk-1 – 4bg–2 %3D Prove that for all…

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Q: So this is the pitcure of the quesiton I did with all the work done. I *think* the area is correct,…

A: We solve this problem by rearranging the triangles in octagon.

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

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Q: (a) Show that if x E Q is such that |x|, < 1 for all prime numbers p then x E Z.

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Q: 3. Show that 1 > 0. 4. Show that if a > b and c < 0, then ac < bc.

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Q: et a, b, c be non-zero integers. Suppose that x, y are integers such that ax + by = c. Show that for…

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Q: ssnip

A: Given: B=abcd A=1234 AB=BA AB=1234abcd     =1×a+c×21×b+2×d3×a+4×c3×b+4×d     =a+2cb+2d3a+4c3b+4d

Q: Let a, b e C. (i) Prove that if |a| <1 and |6| < 1, then a - b < 1. 1- āb

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Q: *3. Suppose x is a real number and > 0. Prove that (x - e, x + e) is a neighborhood of each of its…

A: Given that 'x' is a real number and ε&gt;0. To prove that x-ε, x+ε is a neighborhood of each of its…

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

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Q: An integer x is odd if x2 is odd. Let x, y and z be integers. If x y and y z, then x | z. For any…

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Q: a. Let, a and b be integers such that 1 ≤ a ≤ b. If b = aq + r, where 0 ≤ r ≤ a, then prove that,…

A: What is Greatest Common Divisor: For two integers, their greatest common divisor is an integer that…

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: eld of subsets of 2 and suppose Q: B [0, 1] is a set dditive on B. 1 for all A e B and Q(2) = 1.…

A: Complete proof is provided in step2.

Q: Prove that if A={- :n EN, then A' ={0}. Hint: (a) if x1, then . (d) if x=0, then x EN.

A: A set A=1n:n∈N then A'=0. i.e. the set of limit point contains only 0 element.

Q: (a) Determine the set of values of n for which and are positive integers. A 'Pythagorean triple' is…

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Q: Prove: Let a,b,x and y be integers. If a|x and b|y then (ab)|(2xy).

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Q: 1. Suppose a and b are real numbers. Prove that if 0< a < b then 1/b < 1/a

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Q: Let A, B be sets of real numbers. We will say that B is dense in A if for any s e A and for any real…

A: Given that A,B,C are sets of real numbers with C is dense in B and B is dense in A. We know that a…

Q: If a and b are two integers such that a = bq +r, with b> 0, then %3D gcd(a, b) # gcd(b,r) gcd(a, b)…

A: Let a = bq + r   … (1)Now,  let d1   = gcd ( a,b) then ⇒d1  | a  and   d1  | b ⇒d1 | a - bq   and d1…

Q: 2. Find U, A, and NE,Ą, where: a. A = {-i, i} for every positive integer i. b. A = {i°, i', i²} for…

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Q: Let (a,b)=1 Show that (a+b, a-b)=1 or 2. When is this GCD 1? When is 2?

A: Proof:It is given that gcd(a,b) = 1. So ax + by = 1.Let gcd(a + b, a – b) = d, then d|( a + b) and…

Q: 4.89. Let a E Z. Prove that if 3 5a, then 3 a.

A: Given, 3 divides 5a.  Then, we can write 5a = n*3, where n is an integer.

Q: Let a and b be positive integers such that ax + by %3D 7 for some integers x and y. Then O…

A: Greatest common divisior:  Let a and b be two integers such that gcd(a,b)=d if  d|a and d|b If…