3.15. Use the Miller-Rabin test on each of the following numbers. In each case, either provide a Miller-Rabin witness for the compositeness of n, or conclude that n is probably prime by providing 10 numbers that are not Miller-Rabin witnesses for n. n (c) n = 294439 (a) n = 1105. (Yes, 5 divides n, but this is just a warm-up exercise!) (b) n = 294409 (d) n = 118901509 (f) n = 118901527 (e) n = 118901521 118915387 Solution to Exercise 3.15. (a) n − 1 = 1104 = 24. 69. 60 = n = 269 = -138 (mod 1105) (mod 1105) (mod 1105) (mod 1105) Thus 1105 is composite. It factors as n = 5.13.17. 22.69 = 259 24-69-324 28-69 1

Help solving this please! Notes to consider: For part 1, A podium with (1st: horse A; 2nd: horse B; 3rd: horse C) is different than a podium with (1st: horse B; 2nd: horse C; 3rd: horse A). We're asking how many ways the 1st 2nd and 3rd horses could be.  Let H = {h_0, h_1, ... h_11} be the 12 horses in the race.  Here are two example ways the "podium" can be arranged: (h_3, h_7, h_2), (h_9, h_0, h_11) For part 2, Assume that horses may not share a placement (i.e. they can't "tie"). "Grand Valor, Seabiscuit and Secretariat do not finish first" means that: Grand Valor did not finish first AND Seabiscuirt did not finish first AND Secretariat did not finish first For Part 3,

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Could you help me with this please Is it necessary to learn about calculus and other things that most of us will never again use in our lives? What does calculus do for you? When do you use calculus in the real world?