Show that if a and b are integers with a ≡ b (mod p) for every prime p, then it must be that a = b

Billy Bob is the sole proprietor of Billy Bob's Smokers and is a fabricator of competition-level, wood-fired smokers. Billy Bob is introducing a home-grade version of his smoker and has a contract with a local distributor to produce and deliver 50 units for sale during the summer. More specifically, Billy Bob is required to deliver the first 6 units in May, the next 10 units in June, 14 more in July, and the remaining 20 in August. The first unit produced for delivery in May required 40 hours of labor and the second unit took 30 hours of labor to complete. Assuming a learning rate can be established based on the first two production units, will Billy Bob be able to meet his contractual obligations? Billy Bob works 160 hours per month. You must show all work used to answer the question. That is, provide the number of labor hours that Billy Bob will consume in each month to produce the number of smokers indicated.

Provide (and round as necessary) all numeric values out to four decimal places. In the “Obligation Met” cells, type: Yes or No

. Let φ : R → S be a ring homomorphism of R onto S.

Prove the following:

J ⊂ S is an ideal of S if and only if φ ^−1 (J) is an ideal of R.

Identify and describe three IT system deficiencies that movement to an integrated enterprise IT architecture overcomes

The diameters, in centimetres, of 60 holes bored in engine castings are measured and the results
are as shown. Draw a histogram depicting these results and hence determine the mean, median
and modal values of the distribution.
Diameters 2.011–2.014 2.016-2.019 2.021-2.024 2.026-2.029 2.031-2.034
Number of
holes
7 16 23 9 5

(3) (a) Show that every two-dimensional subspace of R3 is the kernel of some linear transformation T : R3 → R. [Hint: there are many possible ways to approach this problem. One is to use the following fact, typically introduced in multivariable calculus: for every plane P in R3, there are real numbers a, b, c, d such that a point (x,y,z) belongs to P if and only if it satisfies the equation ax+by+cz = d. You may use this fact without proof here, if you like; note that it considers all planes, not just those through the origin.] (b) Are there any other sets W such that W is the kernel of some linear transformation T : R3 → R? (If not, explain why not; if so, explain why the set or sets you mention can be kernels, and why there are no others.) (c) What possibilities are there for the image im(T) of a linear transformation T : R3 → R? (d) What possibilities are there for the kernel and image of a linear transformation S : R → R3?

Write f(x)=x^4+2x^3+2x+1 as a product of irreducible polynomials, considered as a polynomial in Z3[x], Z5[x], and Z7[x], respectively.

1. 2. Let f(x) be as in the previous exercise. Choose D among the polynomial rings in that exercise, so that the factor ring D/〈f(├ x)〉┤i becomes a field. Find the inverse of x+〈f├ (x)〉┤i in this field.

Prove the following formulas, where u, v, z are complex numbers and z = x +iy.

a. sin(u+v) = sin u cos v + cos u sin v.

b. cos(u+v) + cos u cos v - sin u sin v.

c. sin^2 z + cos^2 z = 1.

d. cos(iy) = cosh y, sin (iy) = i sinh y.

e. cos z = cos x cosh y - i sin x sinh y.

f. sin z = sin x cosh y + i cos x sinh y.

Using the Composite Trapezoidal Rule, with evenly spaced nodes, and n=3, find an approximate value for interval where b=1 and a=0, e^(-x^2)dx. Estimate the error.

Show that a graph without isolated vertices has an Eulerian walk if and only if it is connected and all vertices except at most two have even degree.

 T F If y'  = 3y^(2) + 5y − 2 and y(1) = 0, then lim t→∞ y(t) = −2.

Let p and q be propositions.

(i) Show (p →q) ≡ (p ∧ ¬q) →F

(ii.) Why does this equivalency allow us to use the proof by contradiction technique?

Find the general solution for the equations:

P(x) y"+ xy' - y = 0

a) P(x)= x

b) P(x)= x²

c) P(x) = 1