1. For each of the following pairs of numbers a and b, calculate ged(a,b) and find...
1 For each of the following pairs of numbers a and b, calculate and find integers r and s such ged (a; b) by Eucledian algorithm that gcd(a; b) = ra + sb. ia= 203, b-91 ii a = 21, b=8 2 Prove that for n 2 1,2+2+2+2* +...+2 -2n+1 -2 3 Prove that Vn 2 1,8" -3 is divisible by 5. 4 Prove that + n(n+1) = nnīYn E N where N is the set of all positive integers....
2,3,4,5,6 please 2. Use the Euclidean algorithm to find the following: a gcd(100, 101) b. ged(2482, 7633) 3. Prove that if a = bq+r, then ged(a, b) = ged(b,r). such that sa tb ged(a,b) for the following pairs 4. Use Bézout's theorem to find 8 and a. 33, 44 b. 101, 203 c. 10001, 13422 5. Prove by induction that if p is prime and plaja... An, then pla, for at least one Q. (Hint: use n = 2 as...
PROBLEM 1 For each of the following pairs of integers, use the Euclidean Algorithm to find ged(a,b), and to write gcd(a,b) as a linear combination of a and b, i.e. find integers m and n such that gcd(a,b) = am + bn. (a) a = 36, b = 60. (b) a = 12628, b = 21361. (c) a = 901, b = -935. (d) a = 72, b = 714. (e) a = -36, b = -60.
1. Design an algorithm to find all the non-common elements in two sorted lists of numbers. What is the maximum number of comparisons your algorithm makes if the lengths of the two given lists are m and n, ?respectively 2. Estimate how many times faster it will be to find ged(98765, 56789) by Euclid's algorithm compared with the algorithm based on checking consecutive integers from min{m, n} down to gcd(m, n). 3. For each of the following functions, indicate how...
Use R language to program Problem 1: Greatest Common Divisor (GCD) Please write two functions, g edi ) and gcdr , which both take two integers a, b and calculates their greatest common divisor (GCD) using the Euclidean algorithm gcdi () should do so using iteration while gcdr () should use recursion. Then write a third function, gcd(), which takes two integers a, band an optional third argument nethod which takes a charater string containing either "iterative" or "recursive", with...
Please show question 1 (all parts). Thank you! 1. Using the Euclidean algorithm to find the ged of following pairs. Write down the ged as a linear combination of given pairs (a) 524 and 148 in Z (b)33 + 2r +1 and 2 +1 in Zs[] (c) 3 +2r +1 and 1 n Z[] 2. Compute 42001 in Z5 3. Use principal of induction show that 10" 1 mod 9 4. Show that every odd integer is congruent to 1...
Calculate A+B, and A-B for the following pairs of binary numbers using 2's complement. Choose the N value to be 1 more than the minimum necessary to perform the task. C) 1012 , 10112 D) 101101102,010110112
****python**** Q1. Write a recursive function that returns the sum of all numbers up to and including the given value. This function has to be recursive; you may not use loops! For example: Test Result print('%d : %d' % (5, sum_up_to(5))) 5 : 15 Q2, Write a recursive function that counts the number of odd integers in a given list. This function has to be recursive; you may not use loops! For example: Test Result print('%s : %d' % ([2,...
T'he goal of this problem is to establish the following remarkable result: Bezout's theorern. If a, be Z50, then 3x, y є Z such that gcd(a, b) = ax + by. Here ged(a, b) denotes the greatest common divisor of a and b (i.e. the largest positive integer that divides both a and b). Throughout this problem, we'll use the notation (a) Write down five numbers that live in 2Z +3Z. What's a simpler name for the set 2Z +3Z?...
Applied and Computational Questions 1. Pairs of random numbers (r, y) a. How many different pairs are possible? are generated. X and Y are integers between 0 and 5 inclusive. a random variable W is defined to equal the absolute value of the difference between X b. Suppose and Y. How many distinct values are possible for W? Give the pair of dice is rolled once. 2. Let x represent the number of times a one appears when a probability...