5. Suppose n > 0 Show that if ā is the (multiplicative) inverse of a modulo n then erpn(а)-erph (...
Project 2: The Inverse Modulo n [MOD] Textbook Section: 7.37 Directions: The user will input the modulus п they want to work in ( n > 1 ) along with the integer whose inverse they want to find 0<а). Соmputeе a^f-1}$$by implementing one of the standard algorithms: Euclidean Algorithm, Gauss's Algorithm, Fermat's Little Theorem, Euler's Totient Function, or Chinese
A. Find the multiplicative inverse of 52 mod 77. Your answer should be an integer s in the range from 0 through 76. Check your solution by verifying that 52s mod n = 1. Show that for all integers a, b, and c, if aſb and alc, then a-|bc.
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
QUESTION 1 If modular n = 74, e = 5, what is the multiplicative inverse of e?
20. Congruence Modulo 6. in145 (a) Find several integers that are congruent to 5 modulo 6 and then square each of these integers. (b) For each integer m from Part (20a), determine an integer k so that 0 <k < 6 and m2 = k (mod 6). What do you observe? (c) Based on the work in Part (20b), complete the following conjecture: For each integer m, if m = 5 (mod 6), then .... (d) Complete a know-show table...
11.7 Inverse of an upper triangular matri. Suppose the n × n matrix R is upper triangular and invertible, i.e., its diagonal entries are all nonzero. Show that R1 is also upper triangular. Hint. Use back substitution to solve Rsk-en for k 1, , n, and argue that (sk)i -0 for i > k. 11.7 Inverse of an upper triangular matri. Suppose the n × n matrix R is upper triangular and invertible, i.e., its diagonal entries are all nonzero....
We know that we can reduce the base of an exponent modulo m: a(a mod m)k (mod m). But the same is not true of the exponent itself! That is, we cannot write aa mod m (mod m). This is easily seen to be false in general. Consider, for instance, that 210 mod 3 1 but 210 mod 3 mod 3 21 mod 3-2. The correct law for the exponent is more subtle. We will prove it in steps (a)...
Just number 5 please 4. Show that congruence modulo n is an equivale 5. Consider the unit circle C with equation modulo n is an equivalence relation UI arcle C with equation x2 + y2 = 1 in the plane. Thus C = {(x, y) ERX R: x2 + y2 = 1}. Define a relation on C as follows: For (x, y) EC, antipodal point (-x, -y). In symbols, llows: For (x, y) E C (x, y) is related to...
(1) Consider Z with the addition and multiplication mod 3 as usual. Let R=ZgxZg. Define (a, b)+ (a',b) (aab+and(a,b)((aa-bab +a'b) (a) Show that (R, +) is a commutative ring. b) Show that (1,0) is the identity element for the multiplication. c) Show that the equation 22 hs exactly two solutions in R Bonus Problem) Show that (R, +,.) is a field. (Hint: To find multiplicative inverse, first show that a2 + b2メ0 if (a, b)メ(0.0). Then compute (a, b).(a,-b).) (1)...
Exercise 1. (a) Find the inverse of the matrix 0 0 1/2 A= 01/ 31 1/5 1 0 (b) Let N be a nxn matrix with N2 = 0. Show (I. - N)-1 = IA+N. (Hint: Use the definition of the inverse.)