a) 21 sets { [0], [1], [2], ..., [20] }
b)equivalence class of 0 (I.e. [0] ) = {..., -42, -21, 0, 21, 42, ...}
equivalence class of 1(I.e. [1] ) = {..., -41, -20, 0, 22, 43, ...}
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Using a congruence modulo 21? How many sets in the partition of the integers arising? What...
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...
Problem 2. Find a primitive root for 53. Using this, you can
devise a bijection α from the integers modulo 52 to the nonzero
integers modulo 53 with the property that α(a + b) = α(a)· α(b)
modulo 53. Explain. Does the law of exponents get involved at all?
Note: For this to work right, you can think of integers mod 52 as
{0, 1, 2, . . . , 51} or as any complete system of residues modulo
52,...
(i) Prove that the realtion in Z of congruence modulo p is an equivalence relation. Namesly, show that Rp := {(a,b) € ZxZ:a = 5(p)} is reflexive, symmetric and transitive. (ii) Let pe N be fixed. Show that there are exactly p equivalence classes induced by Rp. (iii) Consider the relation S E N defined as: a Sb if and only if a b( i.e., a divides b). Prove that S is an order relation. In other words, S :=...
Given the values of n below, determine, without exhaustive search, etc., how many integers k there are, with gcd(k, n) = 1, and 1 <= k <= n, such that k has a square root modulo n. Do this for (a) n = 143, (b) n = 286, (c) n = 572, (d) n = 1144, and (e) n = 2288. In each case, determine also phi(n), so as to be able to tell what fraction of reduced residue classes...
Using the Euclidean Algorithm show that gcd (193, 977) Now find integers s, t such that 193s +977t-1, and use this to find the value of a that satisfies the congruence 193a 38 (mod 977)
Using the Euclidean Algorithm show that gcd (193, 977) Now find integers s, t such that 193s +977t-1, and use this to find the value of a that satisfies the congruence 193a 38 (mod 977)
How many classes of solutions are there for each of the following congruences? (a) x2 - 1 = 0 mod (168) (b) x2 + 1 = 0 mod (70) (c) x2 + x + 1 = 0 mod (91) (d) x3 + 1 = 0 mod (140) Please note to show how you got the solutions as well as finding out how many classes of solutions there are for each congruence. Please explain every step so I can understand how...
6. Using the Euclidean Algorithm show that gcd (109, 736) 1 Now find integers s, t such that 109s + 736t 1, and use this to find the value of r that satisfies the congruence 109x 71 (mod 736).
6. Using the Euclidean Algorithm show that gcd (109, 736) 1 Now find integers s, t such that 109s + 736t 1, and use this to find the value of r that satisfies the congruence 109x 71 (mod 736).
Required Information Ch 04 Sec 4 EX OG MAIN - Inverse of a modulo m using the Euclidean Algorithm NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Arrange the steps to find an inverse of a modulo m for each of the following pairs of relatively prime integers using the Euclidean algorithm in the order Ch 04 Sec 4 Ex 06 (d) - Inverse of a modulo m...
How to I solve this using modulo? Problem 1: Develop a C program that reads a file of integers called “numbers.txt” It then prints the occurrence count of each digit from [0-9]. Your code must utilize the following function: void freqFun (int num, int counters[]) In this prototype, num represents the number read from the file and counters represents the set of counters that hold the occurrence frequency of each digit. Digits that do not occur in the file shall...
how many integers from 0 through 999,999 contain the digit 4 exactly twice? how many integers from 1 through 1000000 contain the digits 6 at least once