the integer x such that 5 3 (mod 7) 57 (mod 61) e linear congruence 31
(3) Solve the following linear congruence: 271 = 12 mod 39. (4) Solve the following set of simultaneous linear congruences: 3x = 6 mod 11, x = 5 mod 7 and 2x = 3 mod 15.
(1 point) Find the smallest positive integer solution to the following system of congruence: x = 5 (mod 19) = 2 (mod 5) = 7 (mod 11) x =
1. Solve each linear congruence for all integers x so that 0 sx <m a) 11x 8 (mod 57) b) 14x 3 (mod 231)
39. Suppose that the polynomial congruence f(x)0 (mod 7) has two distinct so- 0 lutions, what are the possible number of solutions of the congruence f(x) (mod 49)? 39. Suppose that the polynomial congruence f(x)0 (mod 7) has two distinct so- 0 lutions, what are the possible number of solutions of the congruence f(x) (mod 49)?
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...
6.29 Theorem. Ifa is an integer and v and n ane natural mumbers such that (a,n) 1, then a)+1a (mod n) Now let's apply these observations to solve actual congrucnces 6.30 Question. Consider the congruence x2 (mod 7). Can you think of an appropriate operation we can apply to bath sides of the congruence that would allow us to "solve" for x? If so, is the walue obiained for x a solution to the original congruence? 6.31 Question. Consider the...
Which of the following equations have solution? justify your answers a) x2 =3 (mod 137 ) this mean x to the 2nd power congruence to 3 mod 13 to the 7 power b) x3 =4 (mod 115.239) this mean x to 3rd power congruence to 4 mod 11 to the 5 th power times 23 to the 9th power c) x7=2(mod49) this mean x to the 7th power congruence to 2 mod 7 to the 2nd power d) x7 =...
1. (Complex Multiplication) Let E : y x3 y23 to this congruence mod p. So for example, #E(Z3) = 3 because we have the solutions (0, 0), (1,0) and (2,0) and no more. - x. Then we can reduce E mod p to get mod p for various primes p. We write #E(Z») for the number of solutions This particular equation has some miraculous explore here patterns we (a) Make a chart that lists p, #E(Zp), and #E(Z) - p...
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...
Find all solutions to the congruence x2+ x+ 1≡0 mod 91. (Hint:factor the modulus, use trial and error to find the solutions modulo the factors, and the CRT to combine the results into solutions to the original equations.)