2.6 Consider all the possible sets of two square roots s, t of 1 (mod 35)...

2.6 Consider all the possible sets of two square roots s, t of 1 (mod 35) where s ≢ t (mod 35) (there are six of them, since addition is commutative (mod 35). For all possible combinations, compute gcd(s + t, 35). Which combinations give you a single prime factor of 35?

2.7 Using CRT notation, show what is going on for all the combinations you considered in #2.6. Explain why gcd(s + t, 35) sometimes gave you a factor, and it sometimes did not.

2.8 Explain how you can make a digital signature that is mathematically equivalent to factoring using the results you considered in this assignment.

Expert Solution

Colby Messinger answered 1 year ago

Given two sets S and T, the direct product of S and T is the set...

Given two sets S and T, the direct product of S and T is the set of ordered pairs S × T = {(s, t)|s ∈ S, t ∈ T}.Let V, W be two vector spaces over F. (a) Prove that V × W is a vector space over F under componentwise addition and scalar multiplication (i.e. if (v1, w1),(v2, w2) ∈ V × W, then (v1, w1) + (v2, w2) = (v1+w1, v2+w2) and a(v, w) = (av, aw)...

Express all solutions to the following equations as sets of integers: (a) 7x ≡ 12 mod...

Express all solutions to the following equations as sets of integers: (a) 7x ≡ 12 mod 13 (b) 10x ≡ 4 mod 6 (c) 6x ≡ 8 mod 12

find all primes p such that 6 is a square mod p? please with clear hand...

find all primes p such that 6 is a square mod p? please with clear hand writing

Make a list of all the possible sets of quantum numbers that an electron in an...

Make a list of all the possible sets of quantum numbers that an electron in an atom can have if n = 4. How many different states with n = 4 are there? Indicate on your list which states are degenerate (i.e. have the same energy as other n = 4 states). Assume that the electron is in a multi-electron atom (i.e. not the Hydrogen atom). Does the total number of states agree with the general rule that the number...

S = {(2,5,3)} and T = {(2,0,5)} are two clusters. Two clusters that S and T...

S = {(2,5,3)} and T = {(2,0,5)} are two clusters. Two clusters that S and T spans are L(S) and L(T) . Is the intersection of L (S) and L (T) a vector space? If yes, find this vector space. If no, explain why there is no vector space.

x^3-34x+12=0 1) list all rational roots that are possible according to the Rational Zero Theorm a)...

x^3-34x+12=0 1) list all rational roots that are possible according to the Rational Zero Theorm a) +-1 b)+-1,+-2,+-3,+-4,+-6,+-12 c)+-12 d) 1,2,3,4,6,12 b)use synthetic division to test several possible rational roots in order to identify one actual root c) use the root from part b to solve the equation the solution set it?

Write all possible sets of quantum numbers for an electron in the 4s subshell.

1. Simplify each of the following sets as much as possible. a) ((? ∪ ??)? ∩...

1. Simplify each of the following sets as much as possible. a) ((? ∪ ??)? ∩ (? ∩ ∅?))? b) (ℤ ∩ ℚ+)? ∩ ℤ? 2. Determine the cardinality of each set: a) ((ℂ − ℝ) ∩ ℕ) ∪ (ℤ+ − ℕ) b) ?(ℤ?? ∩ ℤ??)

7.11 Consider two cases involving parallel flow of dry air at V = 1 m/s, T∞...

7.11 Consider two cases involving parallel flow of dry air at V = 1 m/s, T∞ = 45°C, and atmospheric pressure over an isothermal plate at Ts = 20°C. In the first case, Rex,c = 5 × 105, while in the second case the flow is tripped to a turbulent state at x = 0 m. At what x‐location are the thermal boundary layer thicknesses of the two cases equal? What are the local heat fluxes at this location for...

Averages and variation Consider two data sets A and B. The sets are identical except the...

Averages and variation Consider two data sets A and B. The sets are identical except the high value of the data set B is three times greater than the high value of data set A. (a) How does the median of the two data sets compare? (b) How do the means of the two data sets compare? (c) How do the standard deviations of the two data sets compare? (d) How do the box- and –whisker plots of the two...

Question