ACCA F1
Accountant in
Business
Darya Yurevna
Rozhkova
[email protected]
THE BUSINESS ORGANISATION
2012 ZAO KPMG, a company incorporated under the Laws of the Russian Federation, a subsidiary of KPMG Eu
MATH 124 PROBLEM SET 10 COMMENTS
Kledin and Timothy pointed out a detail which I overlooked in problem 1. We need to require
that r be square free, otherwise the solutions need not be the numberator o
MATH 124 PROBLEM SET 5 COMMENTS
The first two problems didnt pose any problems. For the third, the easiest way to do things was
to look modulo 8, 27, and 25 and then use the Chinese remainder theorem
MATH 124 PROBLEM SET 6 COMMENTS
In problem 2, the following detail tripped up a lot of people. The eavesdropper can find d1 and
d2 such that d1 e1 + d2 e2 = 1 by the Euclidean algorithm. However, d1 o
MATH 124 PROBLEM SET 8 COMMENTS
For problem 2, given a triangle with integer side lengths a, b, and c and a 60 degree angle between
the sides of length a and b, the law of cosines tells us that a2 + b
SECTION PROBLEMS
(1) Assuming you knew the continued fraction expansion of , but could not calculate or remember any other approximation of . How could you find a convergent that approximates
to with
Homework 9: Continued Fractions.
In question 1) you are advised to use a calculator. The questions marked are
optional, i.e. not for credit.
1) (a) Find the first 10 terms in the continued fraction ex
Homework 7: Primality Testing
1) Show that if (a, 561) = 1 show that a560 1 mod 561.
2) Suppose that n is an integer such that an1 1 mod n for all a with (a, n) = 1.
Show that either n is odd or n = 2
DIRICHLET SERIES, GENERATING FUNCTIONS, EULER PRODUCTS
(1) Let SL2 (Z) act on the upper half plane as discussed in class. Show that for g Sl2 (Z), the
following holds for imaginary parts:
Im(gz) =
Im(
SECTION EXAMPLES
(1) Find the continued fraction expansion for
for
x2
5y 2
1+ 5
2 .
For
5
2 .
For
5. What are the solutions
= 1?
(2) Show there are infinitely many solutions to x2 + y 2 = z 4 with x,
MATH 124 PROBLEM SET 7 COMMENTS
The only problem people had was with 4c. Problem 4a, b had the hypothesis that the integer
n satisfied 2n1 = 1 mod n. If this held and n is odd, then part a implied 2n
Homework 12: Dirichlet Series.
1) Find the set of all automorphisms of the quadratic form X 2 + XY + Y 2 .
2) Suppose that d < 0 is square free and 1 mod 4.
a) If d 1 mod 8 show that cprop
(2r ) = 1 i
SECTION EXAMPLES
(1) Let SL2 (Z) act on the upper half plane as discussed in class. Show that for g Sl2 (Z), the
following holds for imaginary parts:
Im(gz) =
Im(Z)
|cz + d|2
if g has bottom row c, d.
LAW AND REGULATION
GOVERNING ACCOUNTING
2012 ZAO KPMG, a company incorporated under the Laws of the Russian Federation, a subsidiary of KPMG Europe LLP, and a member firm of the KPMG network
of indep
1. POLITICAL AND LEGAL
FACTORS
2012 ZAO KPMG, a company incorporated under the Laws of the Russian Federation, a subsidiary of KPMG Europe LLP, and a member firm of the KPMG network
of independent me
Accounting and Finance functions
2012 ZAO KPMG, a company incorporated under the Laws of the Russian Federation, a subsidiary of KPMG Europe LLP, and a member firm of the KPMG network
of independent
STAKEHOLDERS
2012 ZAO KPMG, a company incorporated under the Laws of the Russian Federation, a subsidiary of KPMG Europe LLP, and a member firm of the KPMG network
of independent member firms affilia
Chapter: Political and legal factors
1. This same categorisation of environmental factors is sometimes referred to as
SLEPT analysis.
Which of the following categories is not represented within this a
Homework 1: The Euclidean Algorithm
A question marked with a * is optional, i.e. not for credit.
1) Show by induction on n that
1 + 2 + 3 + . + n = n(n + 1)/2.
Also show that
(1 + 2 + 3 + . + n)2 = 13
MATH 124 PROBLEM SET 11 COMMENTS
Problem 1 was fine.
On problem 2, many people didnt use the method we talked about in class using quadratic
forms and by so doing made a lot of extra work for themselv
MATH 124 PROBLEM SET 2 COMMENTS
Here is a good way to write up problem 1: I claim [ pni ] counts the number of multiples of pi less
than or equal to n. Multiples of pi are of the form api with a 1. To
MATH 124 PROBLEM SET 9 COMMENTS
For problem 5, remember that p = [a0 , . . . , an ] and q = [a1 , . . . , an ]. We assume that 2 q
p1
2 .
Then q 0 = [a0 , . . . , an1 ]. We know that p = [a0 , . . .
MATH 124 PROBLEM SET 4 COMMENTS
(3) This is fairly easy to do by picking a generator. However, a more conceptual way to understand it involves viewing (Z/pZ) as the product of the cyclic groups C2r Cs
Congruences
-If a, b Z, where a > 0, then unique q, r Z, s.t.,
b = qa + r, where 0 r < a
Here q is called the quotient of b divided by a, and r is called the remainder of b divided
by a.
-Given a fixe
Prime Numbers
-An integer p > 1 is called a prime if its only positive divisors are 1 and p;
otherwise it is called composite.
-Prime numbers are important because they are the building blocks of the
Equivalence Class Properties
-Let R be any equivalence relation on the set S. If a, b S, then
a [a]
[a] = [b] iff (a, b) R
[a] [b] = iff (a, b) / R
-The set of congruence classes of integers under the
Equivalence Class
-Let A be a set and R be a relation on A. If R is reflexive, symmetric, and
transitive, then we say that R is an equivalence relation.
-Let R be an equivalence relation on the set A,
Linear Congruences
-Congruences are also useful in expressing some of our earlier results. Recall Bezouts
Identity (Theorem 10) which considered whether integers x and y exist which satisfy a
Linear
D
GCD and Prime Numbers
-If a = 2^6 3^4 and b = 2^3 3^2 5^2, then gcd(a, b) = 2^3 3^2.
-prove that if a|n and b|n, then ab|n using the Unique Factorization Theorem
Let P = p1, p2, . . . , pn denote the
Homework 8: Diophantine Equations.
1) For each of the following equations describe all rational solutions:
2x2 y 2 = 1; 5x2 2y 2 = 1; x2 xy + y 2 = 1; x2 xy + y 2 = 2.
2) Find all triangles with integ
Homework 11: Binary Quadratic Forms.
1) Which of the following binary quadratic forms are equivalent. In each case of
equivalence find a unimodular transformation which exhibits the equivalence.
12X 2
Homework 10: Pells Equation.
Questions marked with a * are optional, i.e. not for credit.
1) Suppose that d Z>0 is not a perfect square. Letr Z satisfy r2 + |r| d.
Let [a0 , ., an ]/[a1 , ., an ] deno
Homework 3: Congruences
Questions marked with a * are optional, i.e. not for credit.
1) Solve the following congruences:
(a) 377x 26 mod 949;
(b) 219x 26 mod 949.
2)Find the second smallest positive i
MATH 124 PROBLEM SET 3 COMMENTS
A general note on writing up calculations: you should write out the calculations you had to do
for techniques that deal with a new topic. For example, on this problem s