3. Use the Euclidean algorithm to find integers (x, y) such that 23x + 65y = 1.
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- The values of x and y should be swapped. Does the following algorithm work?Use the extended Euclidean algorithm to express gcd(144, 89) as a linear combination of 144 and 89.Use the extended Euclidean algorithm to express gcd(252, 356) as a linear combination of 252 and 356.e) Show that if 6 integers are chosen at random, at least two numbers will have the same remainder when divided by 5. Use pigeonhole principle.
- Use the extended Euclidean algorithm to find the greatest common divisor of thegiven numbers and express it as a linear combination of the two numbers.(a) 1001 and 1331(b) 9888 and 6060(c) 12345 and 54321Use the Extended Euclidean Algorithm find x and y such that gcd(798, 111) = 798x + 111yYou have to run Prim's algorithm for the problem defined by adjacency matrix: 1 2 3 4 5 6 7 8 9 1 0 10 9 999 999 17 999 999 999 2 10 10 3 9 11 0 14 4 2 999 999 13 999 14 0 7 999 999 999 999 999 4 999 4 7 0 999 2 8 999 999 567 999 2 999 999 0 6 999 1 999 17 999 999 2 6 0 999 7 999 999 999 999 8 999 999 0 11 4 8 999 13 999 999 1 7 11 0 8 9 999 999 999 999 999 999 4 8 0 1. We started from the vertex vl, so initially we have Y = {v1}: initial nearest 1 2 3 4 5 6 7 8 9 16 1 1 1 1 1 1 1 1 distance -1 10 9 999 999 17 999 999 999 Print out the values stored in the nearest and distance arrays after first iteration of Prim's algorithm. Specify the value of vnear and the next vertex that has to be added to Y Hint: use (copy) the table above to record your answer.
- Euclid’s Algorithm states the following: gcd(a, b) - a and b are integers b ≠ 0 r is the remainder of integer division At each step, the remainder, r, decreases by at least 1 r must eventually become 0 Use Euclid’s Algorithm to calculate gcd(96, 128).let n = 1*3*5*....*197*199 (the product of first 100 odd numbers) find the last 2 digits of ni. 45 i. 654 111. 545 iv. 665 V. 6565 Use nl= 0401 and let n2 = 45. Using Fibonacci series, generate the values of n3 up to n10. Do this for i., ii, iv and v.
- 2. There is a more efficient algorithm (in terms of the number of multiplications and additions used) for evaluating polynomials than the conventional algorithm described in the previous exercise. It is called Horner's method. This pseudocode shows how to use this method to find the value of anx" + an-1xn ++ a₁x + ao = 0 at x = c. procedure Horner(c, ao, a₁, a2,..., an: real numbers) y := an for i:=1 to n y := y*c+an-i return y(y = anc" +an-1c"1+...+ a₁c + ao} 2.a Evaluate 3x² + x + 1 at x = 2 by working through each step of the algorithm showing the values assigned at each assignment step. 2.b Exactly how many multiplications and additions are used by this algorithm to evaluate a polynomial of degree n at x = c? (Do not count additions used to increment the loop variable.)x is congruent to 21 mod 8. Then x isLet a and b be integers. Then a divides b if and only if: O ab-k for som integer k you can't divide integers O a=bk for some integer k O b=ak for some integer k