7. Giver an example such that ΣΙΣm-1 m=1 Zn=1 Anm n=1 Zm=1 Unm. 7. Giver an...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
1. Use mathematical induction to prove ZM-1), in Ik + 6 for integers n and k where 1 <k<n - 1. = 2. Show that I" - P(m + k,m) = P(m+n,m+1) (m + 1) F. (You may use any of the formulas (1) through (14”).)
RINGS AND FIELDS
(1.) Show that the ideals in Zm are precisely the set of the form < d> where d divides m
(1.) Show that the ideals in Zm are precisely the set of the form where d divides m
Problem 2. Let me N so that Zm has a primitive root. Show that it has exactly oſo(m)) primitive roots. Hint: Which powers of the primitive root are primitive roots?
3) Find the torque Experienced by a maguchi Menopile giver the following Conditions. a) M = 32 A. m² ² , 3 = 0.25 Tŷ b) u = 0,7 A.M? (-) 3 = 14T (7) c) 4 = 8 Am? }} 3 = 1.6T (-²)
We toss a fair coin n 400 times and denote Zn the number of heads. (a) What are E(Zn) et Var(Zn)1? (b) What is the probability that Z 200? (use the normal approximation together with the continuity correction (c) What is the smallest integer m such that Pr 200-mくZ.く200 +m] > 20%? (use the normal approximation together with the continuity correction).
We toss a fair coin n 400 times and denote Zn the number of heads. (a) What are E(Zn)...
10.) Consider (Zn; n), the set Zn with mod n
multiplication.
i. Argue that if neither a nor b has any common divisors greater
than 1
with n then neither does ab. [Equivalently gcd(a; n) = 1,
etc.]
ii. Argue that if a does not have any common divisors greater than
1
with n, then [a]n has a multiplicative inverse in Zn.
iii. Argue that (i) and (ii) imply that the set of elements
f[a]n 2 Znjgcd(a; n) = 1g...
Need to find the Reaction Quotient for the following: 7.Zn|Zn2+(1.0 M)||Ag+(0.10 M)|Ag 8.Zn|Zn2+(0.10 M)||Ag+(0.10 M)|Ag 12.Pb|Pb2+(1.0 M)||Ag+(0.10 M)|Ag 14.Cu|Cu2+(1.0 M)||Ag+(0.10 M)|Ag
2. [7 marks] (a) Use the semi-empirical binding energy formula (above) and B to obtain the semi-empirical Zm, + Nm, mass formula (SEMF). [Assume A is odd so that the pairing term is zero.] (b) Mirror nuclei are nuclei with the same mass number A and interchanged values of N and Z, e.g. IN-Z] 1. Which terms in the above formula give rise to differences in atomic mass between a pair of mirror nuclei? (c) Calculate the predicted atomic mass...
Let n > 1. Suppose that Zn is cyclic. Prove that n must be prime.