If gcd(a, b) = 1 and c\a, then gcd(c,b) = 1. If d # 1 is the smallest positive divisor of an integer n, then d is prime.

Q: Let n ∈ Z. Prove that if n is divisible by 3, then n = 6k or n = 6k + 3 for some integer k.

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Q: 4. In the figure, if AB = AC and BO = OC. Prove that ZB = C .C B

A: a). Given: AB=AC and BO=OC Prove: ∠B=∠C

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

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: (c) 2, 3, 4+ V10 and 4. V10 are irreducible elements of R.

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Q: If ∠A≡∠C in ABCD, then ABCD is cyclic.

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Q: Prove that o(n) d21 d/n and, in particular, show that if n is perfect, then 2. d21 d/n

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Q: Prove that vn if n is divisible by 3 then n2+2 is not divisible by 3

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Q: Prove that for any integer n, n6k - 1 is divisible by 7 if gcd(n, 7) = 1 and k is a positive…

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

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Q: Show that if m ∈ Z+ and a ∈ Z such that gcd(a, m) = 1 and ordma = m − 1, then m is prime.

A: Use the following definitions, to obtain the solution. The Euler phi function for the positive…

Q: Show that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a, c) = 1.

A: Consider, gcd(a, b) =1 and gcd(a, c) = 1. If gcd(a, b) =1 then there exist integers m and n such…

Q: Show that if m e Z+ and a e Z such that gcd(a, m) = 1 and ordma = m – 1, then m is prime.

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Q: Prove, by induction, that n + 3n2 + 2n is divisible by 6, for n 2 0.

A: In this problem I have use mathematical induction method two times please read step by step.

Q: Prove that ?? is generated by the following n-1 transpositions: (1 2), (2 3), (3 4),. . . , (n-1 n)

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Q: Prove that if r <s < n, then P(n , s) is divisible by P(n,r).

A: Given: r≤s≤n Therefore, n-r≥n-s So we can write n-r! = n-rn-r-1n-r-2...n-sn-s-1n-s-2...1 n-r! =…

Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

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Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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Q: If gcd(a, b) = 1 and c|a, then gcd(c, b) = 1. If d + 1 is the smallest positive divisor of an…

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Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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

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Q: Prove that for any n E N, nº + 2n is divisible by 3.

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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: Prove that if n divides a, b, and c, then n³ divides a²(b – c) + b² (a – c) +²(b – a).

A: A product is a multiplication of two or more terms and these terms are called the factors of the…

Q: Prove that if r < s < n, then P(n, s) is divisible by P(n.r).

A: Use the formula of permutation

Q: 3. Suppose a, nEZ and 0 < a< n. Prove that - u 1 + a a (:) in terms of factorials; and secondly by…

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Q: Show that n is composite if and only if p(n) <n – Vn.

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Q: 1. Let x E Z. Prove that if x4–1 is not divisible by 4, then x2 – 4.x– 3 is also not divisible by 4.

A: As per our guidelines we are supposed to answer only one question.   So, here is the ques. no. 1

Q: Prove that ∀n ∈Z, gcd(7n + 1, 15n + 2) = 1 *use only the definition of divisibility and gcd please!

A: we want to prove that   gcd(7n+1 , 15n+2)=1

Q: Prove that if c is a natural number then for all a and b we have : gcd(ac, bc) = c gcd(a, b)

A: We have to prove that if c is a natural number then for all a and b gcd(ac, bc)=c gcd(a, b) .

Q: Prove that n² +n is divisible by 2, where n E N.

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Q: Prove that Vn if n is divisible by 3 then n2+2 is not divisible by 3

A: If a and b are two integers. Suppose n is non zero integer such that |n|≠1. If a is divisible by n…

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

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: Suppose a is divisible by b. Show that gcd(a, b) = b.

A: Consider the given information. As we have given that a is divisible by b. then there is a number c…

Q: Prove that for all n E N, 7" +2 is divisible by 3.

A: We will use mathematical induction to prove the result in question 4.

Q: If gcd(a, b) = d, then a and- are not relatively prime. d %3D

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Q: а b If gcd(a, b) = d, then d and are not relatively prime. d

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Q: Prove that ℤ/n has exactly n elements.

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Q: P

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Q: 5. Prove that for any integer n, at least one of the integers n, n+2, n +4 is divisible by 3.

A: According to guidelines we can solved only ist question thank you   we have to Prove that for any…

Q: Suppose that A = {1,2,3,4} and P(A)is the powerset of A. Then |PKA)| =..?

A: We will use the formula.

Q: 3. Prove that if p is a prime >3 then the numerator of 1 + 1/2+1/3+...+V/(p-1) is divisible by…

A: Given: Let us consider the given, r=1+12+13+...+1p-1

Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

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: Prove that ∀n ∈ Z, gcd(7n + 1, 15n + 2) = 1.

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