1125 and b 56 (a) Find ged(a, b) using: (i) The Euclidean Algorithm (ii) The fundamental...
Using the Extended Euclidean Algorithm, find the multiplicative inverse of: 31 mod 3480
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...
Foundations of matematics question need help solving. Q1. Consider the Diophantine equation (i). Use Euclid's Algorithm to compute ged(17,60) (ii). Determine the solvability of the Diophantine equation (iii). Use Euclidean algorithm's back substitution to find an ordered pair such that (iv). Find all solutions of the Diophantine equation (v). Find the inverse of 17 modulo 60 01. Consider the Diophantine equation 17x +60y-3 (D. Use Euclid's Algorithm to compute gcd(17,60) (i). Determine the solvability of the Diophantine equation (ii). Use...
a Find the greatest common divisor (gcd) of 322 and 196 by using the Euclidean Algorithm. gcd- By working back in the Euclidean Algorithm, express the gcd in the form 322m196n where m and n are integers b) c) Decide which of the following equations have integer solutions. (i) 322z +196y 42 ii) 322z +196y-57
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,...
B1 a. Let x := 3C1 + 1 and let y := 5C2 + 1. Use the Euclidean algorithm to determine the GCD (x, y), and we denote this integer by g. b. Reverse the steps in this algorithm to find integers a and b with ax+by = 8. c. Use this to find the inverse of x modulo y. If the inverse doesn't exist why not?
Q 9. In this question, you must clearly set out your working, describing your process with full English sentences. (a) Using Euclid's Algorithm 14.13, show that ged(25, 33) = 1, and hence find integers x, y E Z such that x · 25+ y: 33 = 1. Clearly state the values of x and y. (b) By the Multiplicative Inverse Theorem 14.22, it follows from (a) that 25 has a multiplicative inverse, 25(-1), in Z33. Calculate 25(-1) using your answer...
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.
(3) Hint: Use the Euclidean Algorithm (repeated application of division algorithm using previous remain- ders) to find the greatest common divisor of the given pairs of elements and use that to express these principal ideals. (a) Express the ideals as 2178Z2808Z and 2178Zn 2808Z in Z as principal ideals. (b) Express the ideals (2r63 r+2x + 2) + (2r5 +3x4 + 4x +x+ 4) and (2r63z4 as principal ideals +2x +2)n (2r5 +34 + 4x3 + z2+4) in (Z/5Z) (3)...
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...