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Let n be a positive integer. For each possible pair i, j of integers with 1<i<i...
Let n be a positive integer. For each possible pair i, j of integers with 1...
Let n be a positive integer. For each possible pair i, j of integers with 1 sisi<n, find an n x n matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
(6) Let A denote an m x n matrix. Prove that rank A < 1 if...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
1. Let n be a positive integer with n > 1000. Prove that n is divisible...
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.
Problem 3 Let n and k > l be positive integers. How many different integer solutions...
Problem 3 Let n and k > l be positive integers. How many different integer solutions are there to x1 +...+ In = k, with all xi <l?
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer,...
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer, and let m = 6q+r where q and r are integers with 0 <r < 6. Use (a) and rules of exponents to show that 2" = 2 mod 9 (c) Use (b) to find an s in {0,1,...,8} with 21024 = s mod 9.
7. (10) Given an array of integers A[1..n], such that, for all i, 1 <i< n,...
7. (10) Given an array of integers A[1..n], such that, for all i, 1 <i< n, we have |Ali]- Ali+1]| < 1. Let A[1] = and Alny such that r < y. Using the divide-and-conquer technique, describe in English algorithm to find j such that Alj] z for a given value z, xz < y. Show that your algorithm's running time is o(n) and that it is correct o(n) search an 2 8. (10) Solve the recurrence in asymptotically tight...
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a)...
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
Prove each of the following statements is true for all positive integers using mathematical induction. Please...
Prove each of the following statements is true for all positive integers using mathematical induction. Please utilize the structure, steps, and terminology demonstrated in class. 5. n!<n"
(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random...
(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random variables {Xn} with pdf, fx, (x) = xht where 1<x<. Obtain Fx (2) and hence find the limiting distribution of X, as noo. c) Consider a random sample of size n from Fx (x) = where - <I<0. Find the limiting distribution of Yn as n + if (a)' = n max{X1, X2, X3,...,xn). and X(n) [17 marks]
Let n be a positive integer. For each possible pair i, j of integers with 1 sisi<n, find an n x n matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.
Problem 3 Let n and k > l be positive integers. How many different integer solutions are there to x1 +...+ In = k, with all xi <l?
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer, and let m = 6q+r where q and r are integers with 0 <r < 6. Use (a) and rules of exponents to show that 2" = 2 mod 9 (c) Use (b) to find an s in {0,1,...,8} with 21024 = s mod 9.
7. (10) Given an array of integers A[1..n], such that, for all i, 1 <i< n, we have |Ali]- Ali+1]| < 1. Let A[1] = and Alny such that r < y. Using the divide-and-conquer technique, describe in English algorithm to find j such that Alj] z for a given value z, xz < y. Show that your algorithm's running time is o(n) and that it is correct o(n) search an 2 8. (10) Solve the recurrence in asymptotically tight...
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
Prove each of the following statements is true for all positive integers using mathematical induction. Please utilize the structure, steps, and terminology demonstrated in class. 5. n!<n"
(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random variables {Xn} with pdf, fx, (x) = xht where 1<x<. Obtain Fx (2) and hence find the limiting distribution of X, as noo. c) Consider a random sample of size n from Fx (x) = where - <I<0. Find the limiting distribution of Yn as n + if (a)' = n max{X1, X2, X3,...,xn). and X(n) [17 marks]
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