Solution
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Let Xe be the set of integers x which satisfy the system of congruences 42 mod 3121, 7 mod 11, c ...
5. (20 points) Solve the system of congruences x (mod 13) and 11 (mod 24). Find the smallest nonnegative integer solution to the system.
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to find 712 mod 13 11. What sequence of pseudorandom numbers is generated using the linear congruential generator Xn+1 (4xn + 1) mod 7 with seed xo 3? 9. Use the construction in the proof of the Chinese...
Arrange the steps in the correct order to solve the system of congruences x 2 (mod 3), x 1 mod 4). and x3 (mod 5) using the method of back substitution Rank the options below Thus, x= 31.2 - 3/4 + 1)2 - 120+5 We substitute this into the third congruence to obtain 12.5 13 mod 5), which implesu li imod 5) Hence, w5v4 and so x 12.5 - 12/5 + 4) - 5 - 60v. 53, where vis an...
3. (16 points) Solve the system of linear congruences using the Chinese Remainder Theorem. 4 (mod 11) a 11 (mod 12) x=0 (mod 13) b. (6 pts) Find the inverses n (mod 11), n21 (mod 12), and nz1 (mod 13). Using these ingredients find the common solution a (mod N) to the system. c. (4 pts) 4. (8 points) What is 1!+ 23+50! congruent to modulo 14?
Problem 11.16. Let X = {XE Ζ+ : x-100): that is, X is the set of all integers from l to 100. For each Y E 9(X) we define AY (2 E 9(X) : Y and Z have the same number of elements) (a) Prove that AY : Y є 9(X)} partitions 9(X). (b) Letdenote the equivalence relation on (X) that is associated with this partition (according to Theorem 11.4). If possible, find A, B, and C such that 1....
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
1. Let C be the set of points which satisfy y. Write down an integral which finds the distance from (1, 1) to (2,22/5) along the curve. Do not directly evaluate this integral. Instead use Simpson's rule with n 4 to approximate the length. 1. Let C be the set of points which satisfy y. Write down an integral which finds the distance from (1, 1) to (2,22/5) along the curve. Do not directly evaluate this integral. Instead use Simpson's...
(3) Let L (a20- 0). Show that the set of functions which satisfy L(u) form an affine lincar subspace. = g(x,t) (3) Let L (a20- 0). Show that the set of functions which satisfy L(u) form an affine lincar subspace. = g(x,t)
B3 a. Solve for x in this equation: 2x + 11 = 2 (mod 4). b. What are the sets of units and zero divisors in the ring of integers modulo 22? (Specify at least the smaller set using set-roster notation.) c. Find a formula for the quotient and the exact remainder when 534 is divided by 8. Hint: find the remainder first by modular arithmetic. Then subtract the remainder from the power and divide to find the quotient.
In C program #include<stdio.h> The first 11 prime integers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. A positive integer between 1 and 1000 (inclusive), other than the first 11 prime integers, is prime if it is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. Write a program that prompts the user to enter a positive integer between 1 and 1000 (inclusive) and that outputs whether the number...