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Mathematics - Algebraic Number Theory

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Marriage penalty eliminated.
The value of the expression 4220 + 0.25(x - 30,650)is the 2006 federal income tax for a single taxpayer with taxable income of x dollars, where x is over $30,650 but not over $74,200.
a) Simplify the expression.
b) Find the amount of tax for a single taxpayer with taxable income of $40,000.
c) Who pays more, a married couple with a joint taxable income of $80,000 or two single taxpayers with taxable incomes of $40,000 each?

I've come up with 2 solutions and not sure which one is correct please help..

a) Simplify the expression.
4220 + .25x - .25*30650
4220+.25x - 7226.5
.25x + -3442.5

a) 4220 + 0.25(x - 30,650)
4220 + (.25x - .25(30,650))
4200 + (.25x - 7,662.50)

b)4220+ 40000/4 - 30650/4
4220+10000-7662.5
4220 + (2,337.5) = $6,557.50

c) THIS IS THE ONE GIVING ME THE PROBLEM I'M GETTING TO SOLUTIONS..
8440 + 8000/4-61300/4
8440 + 2000 - 15325
13115
2*6557.5 = 13115

They pay the same, $13,115
4220 + (.25(80,000) - 7,662.5)
4220 + (20,000 - 7,662.5)
4220+ (12337.5) = $16,557.5

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Solution Preview

(a) Tax = 4220 + 0.25(x - 30650)

= 4220 + 0.25x - 0.25 * 30650

= 0.25x - 3442.50

(b) Tax = ...

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Fundamental Math: Algebraic Number Theory

See the attached file.

1) Compute the Cayley tables for the additive group Z and for the multiplicative group
Z of non-zero elements in Z.

2) Let G be a group written additively. Recall that the order of an element a is the
minimal natural number n such that na = 0. If such n does not exits then one says that
the order of a is infinity.

i) Find the order of the following elements 2; 3; 5; 6 2 Z12.

ii) If G is a group written multiplicatively, the order of an element a is the minimal
natural number n such that an = 1. Find the order of the elements

iii) Find the order of the following elements...

3) i) Let G be a group written additively. An element a of a group G is called a generator
if any element x 2 G has the form x = na for some integer n. For example ????1 and 1 are
generators of Z, while Q has no generators at all. Find all generators of the group Z12.
ii) In multiplicative notation, an element a of a group G is called a generator if any
element of G can be written as a power of a. Carl Friedrich Gauss proved that for any
prime p the group Zp has a generator. Verify this statement for all primes 17 giving
explicitly a generator of the group Zp in each case.

Remark. Can you see any regularity among these generators for di erent primes? Probably not. A conjecture of Artin (which is still open) claims that if a is an integer which is
not a perfect square there are in nitely many primes p for which a is a generator in Zp.

4) i) Let G be a group written multiplicatively. For any element a 2 G, consider the
map fa : G ! G given by fa(x) = ax. Prove that fa is always a bijection.

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