Let n, m, a, b be positive integers. motl Lof oontredietion u th

Q: The generators of Z,, are: (1,57,811) O (13,7.11) O ass711) 'O 05711) O O O O

A: we have to find generators of z12

Q: &2/ suppost se =20, 1} and c={208{ fie ld =20,18 and e= Enumerate the class of all 6- Containing C…

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Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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Q: (2) Prove that 21+ + for all 21, 22 € C.

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Q: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

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Q: Prove dis prove that the or smallest non- commutative ving is of order 4-

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Q: If z is an odd ratural number then 2^+1 is divisi ble by 3 hìnt: prove by Induction

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Q: If lal = n, show that |a'| %3D gcd(n,t)

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Q: Show that there exist no two positive integers nandr(r<n)such that "C, "C,+1 "C+2 and "C,.3 are in…

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Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

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Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L,=u, + 2u,-1

A: Introduction: We know that the relation of Fibocanci number and Lucas numbers is given by  F0=0,…

Q: Prove that multiplication is associative in Q.

A: Prove that multiplication is associative in Q. Let r, s, t be in Q. Then  r=ad, s=be, t=cf; such…

Q: Let A be the set of positive integers that are less than or equal to 40 and are multiples of 4 and…

A: Given: Let A be the set of positive integers that are less than or equal to 40 and are multiples of…

Q: 4 Prove that סלא is Riomonn inkgpable

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Q: show

A: Given : m+n and n+p are odd  So, we can write, m+n = 2s+1 and, n+p = 2t+1, where, s &amp; t are…

Q: Show that if a1,: a2., an are integers that are not all 0 and c is a positive integers, then…

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Q: 2. Prove that L(cashkt)= ki 52-K2

A: To prove that  L(cosh(kt))=ss2-k2

Q: Determine the identity element of the set N={1,2,3,..} of positive integers with addition.

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Q: A cosets of of 212 are اخترأحد الخیارات + 2, +1,.1 O + 1,.2 O + 4, + 1, .3 + 4, + 2, + 1 , .4 O

A: Cosets Definition: Let G be a group and let H is a nonempty subset of G. A left coset of H in G is a…

Q: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…

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Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: 23. Find de V100- tan O coton 2 Opbon 1

A: We have the given integral as ∫dx100-9x2 On solving the given integral furthermore, we get the…

Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: #3. Let n be an mteg er with nz1 and Let b., bzba be ... positive real nmmbers, not all egual, such…

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Q: If a and b are relatively prime then alb

A: Given below the detailed solution

Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: O Prove that G(Vatienal number bers)is negrigibleset

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Q: Let a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: Suppose that lal = 12. The number of distinct left cosets of in is: * %3D 12 O 10 O None of the…

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Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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Q: Prove that if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  =&gt; k | {(n+4) - (n-1)} =&gt; k | 5

Q: Let z, y, z be integers. Prove that if z +y + zis odd, at least one of z, y, or z is odd.

A: We prove this by contradiction

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

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Q: Let i be a positive integer and A, = {..., i – 6, i - 4, i – 2, i, i + 2, i+ 4, i + 6, ... }. Then…

A: First option is correct

Q: Let i be a positive integer and A = [-i + 1 ,1]. Then UA,

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Q: Let Up be avoriable that 8 tand foor ang f th element of the population 3, 7, 8 and Ug be a woriable…

A: It is given that Population U1 : 3, 7, 8 Population U2 : 2, 4 NOTE : According to our guidelines we…

Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

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Q: [Hint: It suffices to show that the numbe 11. Verify that 0, 1, 2, 2², 2³, . , 2° form a complete…

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Q: Prove that tim 51

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 2. Show directly that sup(4) = 1 for A = {1- : n eN}.

A: As we know that, 1&gt;1-1n ∀ n∈ℕ Therefore 1 is the upper bound of the set 1-1n:n∈ℕ. And for each…

Q: 9:Let S [o,1] Prove that inf s=o

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Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Q: prove that, it ņ is an 3n+2is oald then integer and nis add

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