Suppose a, n E Z and 0 < a

Q: .. with n, m, E No for all i e {1,...,r}. Let a = p p... p and b p P (The existence of such a…

A: Let a=p1n1.p2n2.....prnr and b=p1m1.p2m2.....prmr with ni,mi∈ℕ0 For all i∈{1,.....,r}. To prove…

Q: 2. Let S be the set [9] = {1.2..9}. 1 (D) How many derangements of S have 9 in the 2nd position and…

A: We have the set 9=1,2,...,9 We need to find the number of derangements. There are 9 places to fill…

Q: The permutations p= p and q in Se have the two-line symbols 1 2 3 4 5 6 4 621 5 3 and q= 1 2345 1 5…

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Q: 1. Let n be a positive integer and le {1,...,n} such that n divides a, (ai+1-1) for i =

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Q: (9) Let n and k be positive integers. Recall from lecture that a weak composition of n into k parts…

A: 9.For  be positive integers, a weak composition of  into  parts is a -tuple  of nonnegative integers…

Q: How many k-lists {r1, x2, ..., xk} are possible, such that each x; is a positive integer and 1 <x1 <…

A: Here each xi is a positive integer and 1≤x1≤x2≤...≤xk≤n.

Q: Births of boys with n = 20 births and p = 0.512 for a boy. In P(fewer than 8 boys)

A: Given, n=20 p=0.512 Normal as approximation to binomial will be given by as follows: X~Nnp,…

Q: (123456 Let o = find o25 314562 ect one: 123456 612 435) (123456 124365 (123456 314562 (123456…

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Q: Prove that for every n E Z, n² – n is even. (Proof by cases)

A: In the given question we have to prove that for every n∈ℤ, n2-n is even.

Q: Consider the set A={ Identify An B. 3n+2 4n-1 4n+1 {In € N*}. 5n-3 In € N*} and B={

A: Kindly refer to Step 2 for the solution.

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: The factorial of a positive integer n, denoted n!, is equal to the product of all integers between 1…

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Q: Prove that for all n € N, 91(4" + 6n + 8) and n € N, 31(n³-n).

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Q: Prove that, for n = 1, 2, ..., we have 1 1 + 1.2 2.3 +...+ n(n+1) = n n+1

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

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Q: Prove that n < 2n for all n ∈N

A: We prove n&lt;2n by principle of mathematical induction Principle of mathematical inductionStep 1:…

Q: Prove that µ(d) p(n) = n . P ulp

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Q: 4g - 2x2 + 5% -1- Σακ(π)κ. 4x4 k>0 b) -(m)2 + (r)1 + 6 = Σb k>0

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Q: Let m%3= i=1 where pi 0 are integers then i=1 i=1 gcd(m, n) = I min(ai,bi). %3D i=1

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

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: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Prove that for all n in J, 1 + 3 + 5 + • • • + (2n - 1) = n2

A: We have to prove 1+3+5+....+2n-1=n2

Q: Remark: For positive integers n, m, we have • (n) n (m) (n) + (m) = = (lcm (n, m)) (gcd (n, m))

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Q: 1. Let o(n) = Edn d. Example: For primes p, q, a(p) = p+ 1. a(pq) = pq+ p+q+ 1. Show that o(n) = n…

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Q: 1 + 3-1 V3+1 = q +3 V3 %3D

A: P=5, q=2

Q: Eratosthene5-Legendre

A: It is problem of analytic number theory

Q: et p(k) be the number of permutations of (1,2,3,4,5,6) such that there are actly k numbers remain as…

A: A fixed point in a permutation is an element that remains in its original position after the…

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Prove that, for all integers n ≥ 0 and 0 ≤ k ≤n, we have (x+¹)=()+(*+¹) ++ (1) n+ k+1 k Hint: the…

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