A Concise Introduction To Pure Mathematics 4th Edition Martin Liebeck - Solutions

(a) Encode the message WHERE ARE YOU using the public key \((N, e)=\) \((143,11)\).(b) You intercept the encoded answer. Here it is:\(12,59,14,114,59,14\).Brilliantly crack this code and decipher this message.
Critic Ivor Smallbrain has been given the honour of being given a chance to nominate the best film of all time by the Oscar committee. He has to send his nomination to them using the RSA code with public key \((N, e)=(1081,25)\). The nomination will only be accepted if it remains secret until the
Evaluate the binomial coefficients \(\binom{8}{3}\) and \(\binom{15}{5}\).
Liebeck, Einstein and Hawking pinch their jokes from a joke book which contains 12 jokes. Each year Liebeck tells six jokes, Einstein tells four and Hawking tells two (and everyone tells different jokes). For how many years can they go on, never telling the same three sets of jokes?
(a) How many solutions are there of the equation \(x+y+z+t=14\), where \(x, y, z, t\) are non-negative integers? (b) How many solutions are there of the equation \(x+y+z+t=14\), where \(x, y, z, t\) are positive integers and \(t \leq 8\) ?(c) Let \(c_{1}, \ldots, c_{r}\) be integers, and let \(N\)
Josephine lives in the lovely city of Blockville. Every day Josephine walks from her home to Blockville High School, which is located 10 blocks east and 14 blocks north from home. She always takes a shortest walk of 24 blocks.(a) How many different walks are possible?(b) 4 blocks east and 5 blocks
(a) How many words of ten or fewer letters can be formed using the alphabet \(\{a, b\}\) ?(b) Using the alphabet \(\{a,b, c,d, e, f\}\), how many six letter words are there that use all six letters, in which no two of the letters \(a,b, c\) occur consecutively?
(a) Find the number of arrangements of the set \(\{1,2, \ldots, n\}\) in which the numbers 1,2 appear as neighbours.(b) Let \(n \geq 5\). Find the number of arrangements of the set \(\{1,2, \ldots, n\}\) in which the numbers \(1,2,3\) appear as neighbours in order, and so do the numbers 4,5 .
Liebeck has \(n\) steaks and is surrounded by \(n\) hungry wolves. He throws each of the steaks to a random wolf. What is the chance that(i) every wolf gets a steak?(ii) exactly one wolf does not get a steak?(iii) Liebeck gets eaten, in the case where \(n=7\) ?
(a) Prove that\[ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1} \](b) Prove that for any positive integer \(n\),\[ 3^{n}=\sum_{k=0}^{n}\binom{n}{k} 2^{k} \]
\(n\) points are placed on a circle, and each pair of points is joined by a straight line. The points are chosen so that no three of these lines pass through the same point. Let \(r_{n}\) be the number of regions into which the interior of the circle is divided.Draw pictures to calculate \(r_{n}\)
The digits \(1,2,3,4,5,6\) are written down in some order to form a sixdigit number.(a) How many such six-digit numbers are there altogether?(b) How many such numbers are even?(c) How many are divisible by 4 ?(d) How many are divisible by 8 ? (Hint: First show that the remainder on dividing a
(a) Find the coefficient of \(x^{15}\) in \((1+x)^{18}\).(b) Find the coefficient of \(x^{4}\) in \(\left(2 x^{3}-\frac{1}{x^{2}}\right)^{8}\).(c) Find the constant term in the expansion of \(\left(y+x^{2}-\frac{1}{x y}\right)^{10}\).
The rules of a lottery are as follows: You select 10 numbers between 1 and 50. On lottery night, celebrity mathematician Richard Thomas chooses at random 6 "correct" numbers. If your 10 numbers include all 6 correct ones, you win.Work out your chance of winning the lottery.
Here's another way to prove Fermat's Little Theorem. Let \(p\) be a prime number.(a) Show that if \(r, s\) are positive integers such that \(s\) divides \(r, p\) divides \(r\) and \(p\) does not divide \(s\), then \(p\) divides \(\frac{r}{s}\).(b) Deduce that \(p\) divides the binomial coefficient
At a party with six rather decisive people, any two people either like each other or dislike each other. Prove that at this party, either (i) there are three people all of whom like each other, or (ii) there are three people, all of whom dislike each other.Show that it is possible to have a party
Prove that if \(r_{n}\) is as in Question 10, then for any \(n\),\[ r_{n}=1+\binom{n}{2}+\binom{n}{4} \]Was your conjecture in Question 10 correct?
The other day, critic Ivor Smallbrain gave a lecture to an audience consisting of five mathematicians. Each mathematician fell asleep exactly twice during the lecture. For each pair of mathematicians, there was a moment during the lecture when they were both asleep. Prove that there was a moment
(a) Let \(A, B\) be sets. Prove that \(A \cup B=A\) if and only if \(B \subseteq A\).(b) Prove that \((A-C) \cap(B-C)=(A \cap B)-C\) for all sets \(A, B, C\).
Which of the following statements are true and which are false? Give proofs or counterexamples.(a) For any sets \(A, B, C\), we have\[ A \cup(B \cap C)=(A \cup B) \cap(A \cup C) . \](b) For any sets \(A, B, C\), we have\[ (A-B)-C=A-(B-C) . \](c) For any sets \(A, B, C\), we have\[ (A-B)
Work out \(\bigcup_{n=1}^{\infty} A_{n}\) and \(\bigcap_{n=1}^{\infty} A_{n}\), where \(A_{n}\) is defined as follows for \(n \in \mathbb{N}\) :(a) \(A_{n}=\{x \in \mathbb{R} \mid x>n\}\).(b) \(A_{n}=\left\{x \in \mathbb{R} \left\lvert\, \frac{1}{n}
(a) \(73 \%\) of British people like cheese, \(76 \%\) like apples and \(10 \%\) like neither. What percentage like both cheese and apples?(b) In a class of 30 children, everyone supports at least one of three teams: 16 support Manchester United, 17 support Stoke City and 14 support Doncaster
How many integers between 1 and 10000 are neither squares nor cubes?
How many integers between 2 and 10000 are \(r^{\text {th }}\) powers for some \(r \in\) \(\{2,3,4,5\} ?\)
(a) Find the number of integers between 1 and 5000 that are divisible by neither 3 nor 4 .(b) Find the number of integers between 1 and 5000 that are divisible by none of the numbers 3,4 and 5 .(c) Find the number of integers between 1 and 5000 that are divisible by one or more of the numbers 4,5
(a) The equality\[ \sum_{r=0}^{n}\binom{n}{r}=2^{n} \]is given just after Proposition 16.3. Use this to give an alternative proof of Proposition 17.4.(b) Give yet another proof of Proposition 17.4 by induction on \(n\).
(a) Calculate \(\phi(1000)\) and \(\phi(999)\), where \(\phi\) is Euler's \(\phi\)-function.(b) Find the minimum and maximum values of \(\phi(n)\) for \(20 \leq n \leq 30\).(c) Show that if \(n \geq 3\) then \(\phi(n)\) is even.(d) Find all positive integers \(n\) such that \(\phi(n)\) is not
Prove that if \(m\) and \(n\) are coprime positive integers, then \(\phi(m n)=\phi(m) \phi(n)\).
For a positive integer \(n\), define\[ F(n)=\sum_{d \mid n} \phi(d) \]where the sum is over the positive divisors \(d\) of \(n\), including both 1 and n. (For example, the positive divisors of 15 are \(1,3,5\) and 15.)(a) Calculate \(F(15)\) and \(F(100)\).(b) Calculate \(F\left(p^{r}\right)\),
Let \(n\) be a positive integer, and let \(D(n)\) be the set of arrangements of \(\{1, \ldots, n\}\) for which no number is in its corresponding position. (For example, if \(n=4\) then the arrangement \(4,2,3,1\) is not in \(D(4)\) as the number 2 is in position 2; but the arrangement 4,3,2,1 is in
Some time ago, critic Ivor Smallbrain threw a lavish party for 5 of his best friends, including a private showing of the fabulous new Ally Wooden film Everything You Wanted to Know About Sets But Were Afraid to Ask. It was a rainy day, and each guest brought their own umbrella. At the end of the
(a) Find \(r\) with \(0 \leq r \leq 10\) such that \(7^{137} \equiv r \bmod 11\).(b) Find \(r\) with \(0 \leq r
Let \(p\) be a prime number and \(k\) a positive integer.(a) Show that if \(x\) is an integer such that \(x^{2} \equiv x \bmod p\), then \(x \equiv 0\) or \(1 \bmod p\).(b) Show that if \(x\) is an integer such that \(x^{2} \equiv x \bmod p^{k}\), then \(x \equiv 0\) or \(1 \bmod p^{k}\).
For each of the following congruence equations, either find a solution \(x \in \mathbb{Z}\) or show that no solution exists:(a) \(99 x \equiv 18 \bmod 30\).(b) \(91 x \equiv 84 \bmod 143\).(c) \(x^{2} \equiv 2 \bmod 5\).(d) \(x^{2}+x+1 \equiv 0 \bmod 5\).(e) \(x^{2}+x+1 \equiv 0 \bmod 7\).
(a) Use the fact that 7 divides 1001 to find your own "rule of 7." Use your rule to work out the remainder when 6005004003002001 is divided by 7.(b) 13 also divides 1001 . Use this to get a rule of 13 and find the remainder when 6005004003002001 is divided by 13 .(c) Use the observation that \(27
Let \(p\) be a prime number, and let \(a\) be an integer that is not divisible by \(p\). Prove that the congruence equation \(a x \equiv 1 \bmod p\) has a solution \(x \in \mathbb{Z}\).
Show that every square is congruent to 0,1 or -1 modulo 5 , and is congruent to 0,1 or 4 modulo 8 .Suppose \(n\) is a positive integer such that both \(2 n+1\) and \(3 n+1\) are squares. Prove that \(n\) is divisible by 40 .Find a value of \(n\) such that \(2 n+1\) and \(3 n+1\) are squares. Can
Find \(\bar{x}, \bar{y} \in \mathbb{Z}_{15}\) such that \(\bar{x} \bar{y}=\overline{0}\) but \(\bar{x} eq \overline{0}, \bar{y} eq \overline{0}\).Find a condition on \(m\) such that the equality \(\bar{x} \bar{y}=\overline{0}\) in \(\mathbb{Z}_{m}\) implies that either \(\bar{x}=\overline{0}\) or
Let \(p\) be a prime and let \(\bar{a}, \bar{b} \in \mathbb{Z}_{p}\), with \(\bar{a} eq \overline{0}\) and \(\bar{b} eq \overline{0}\). Prove that the equation \(\bar{a} \bar{x}=\bar{b}\) has a solution for \(\bar{x} \in \mathbb{Z}_{p}\).
Construct the addition and multiplication tables for \(\mathbb{Z}_{6}\). Find all solutions in \(\mathbb{Z}_{6}\) of the equation \(\bar{x}^{2}+\bar{x}=0\).
(a) Find \(3^{301}(\bmod 11), 5^{110}(\bmod 13)\) and \(7^{1388}(\bmod 127)\).(b) Show that \(n^{7}-n\) is divisible by 42 for all positive integers \(n\).
Let \(N=561=3 \cdot 11 \cdot 17\). Show that \(a^{N-1} \equiv 1 \bmod N\) for all integers \(a\) coprime to \(N\).
Let \(p\) be a prime number and \(k\) a positive integer.(a) Show that if \(p\) is odd and \(x\) is an integer such that \(x^{2} \equiv 1 \bmod p^{k}\), then \(x \equiv \pm 1 \bmod p^{k}\).(b) Find the solutions of the congruence equation \(x^{2} \equiv 1 \bmod 2^{k}\). (Hint: There are different
Show that if \(p\) and \(q\) are distinct primes, then \(p^{q-1}+q^{p-1} \equiv 1 \bmod p q\).
The number \((p-1)!(\bmod p)\) came up in our proof of Fermat's Little Theorem, although we didn't need to find it. Calculate \((p-1)!(\bmod p)\) for some small prime numbers \(p\). Find a pattern and make a conjecture. Prove your conjecture!
(a) Solve the congruence equation \(x^{3} \equiv 2 \bmod 29\).(b) Find the \(7^{\text {th }}\) root of 12 modulo 143 (i.e., solve \(x^{7} \equiv 12 \bmod 143\) ).(c) Find the \(11^{\text {th }}\) root of 2 modulo 143 .
In a late-night showing of the Spanish cult movie Teorema Poca de Fermat, critic Ivor Smallbrain is dreaming that the answer to life, the universe and everything is 1387 , provided this number is prime. He tries Fermat's test on 1387, then Miller's test, both with the base 2.What are the results of
Prove Liebeck's triplet prime conjecture: the only triplet of primes of the form \(p, p+2, p+4\) is \(\{3,5,7\}\).
Let \(n\) be an integer with \(n \geq 2\). Suppose that for every prime \(p \leq \sqrt{n}, p\) does not divide \(n\). Prove that \(n\) is prime.Is 221 prime? Is 223 prime?
For a positive integer \(n\), define \(\phi(n)\) to be the number of positive integers \(a
There has been quite a bit of work over the years on trying to find a nice formula that takes many prime values. For example, \(x^{2}+x+41\) is prime for all integers \(x\) such that \(-40 \leq x
On his release from prison, critic Ivor Smallbrain rushes out to see the latest film, Prime and Prejudice. During the film Ivor attempts to think of ten consecutive positive integers, none of which is prime. He fails.Help Ivor by showing that if \(N=11!+2\), then none of the numbers \(N, N+1, N+2,
Find the prime factorization of 111111.
(a) Which positive integers have exactly three positive divisors?(b) Which positive integers have exactly four positive divisors?
Suppose \(n \geq 2\) is an integer with the property that whenever a prime \(p\) divides \(n, p^{2}\) also divides \(n\) (i.e., all primes in the prime factorization of \(n\) appear at least to the power 2). Prove that \(n\) can be written as the product of a square and a cube.
(a) Prove that \(2^{\frac{1}{3}}\) and \(3^{\frac{1}{3}}\) are irrational.(b) Let \(m\) and \(n\) be positive integers. Prove that \(m^{\frac{1}{n}}\) is rational if and only if \(m\) is an \(n^{\text {th }}\) power (i.e., \(m=c^{n}\) for some integer \(c\) ).
(a) Which pairs of positive integers \(m, n\) have \(h c f(m, n)=50\) and \(l c m(m, n)\) \(=1500\) ?(b) Show that if \(m, n\) are positive integers, then \(h c f(m, n)\) divides \(l c m(m, n)\). When does \(h c f(m, n)=l c m(m, n)\) ?(c) Show that if \(m, n\) are positive integers, then there are
Find all solutions \(x, y \in \mathbb{Z}\) to the following Diophantine equations:(a) \(x^{2}=y^{3}\).(b) \(x^{2}-x=y^{3}\).(c) \(x^{2}=y^{4}-77\).(d) \(x^{3}=4 y^{2}+4 y-3\).
Prove that if a connected plane graph has \(v\) vertices and \(e\) edges, and \(v \geq 3\), then \(e \leq 3 v-6\).
Prove that it is impossible to make a football out of exactly 9 squares and \(m\) octagons, where \(m \geq 4\). (In this context, a "football" is a convex polyhedron in which at least 3 edges meet at each vertex.)
Prove that if a finite connected plane graph has no faces, then it has a vertex that is joined to exactly one other vertex.
A train leaves Moscow for St. Petersburg every 7 hours, on the hour. Show that on some days it is possible to catch this train at 9 a.m.Whenever there is a 9 a.m. train, Ivan takes it to visit his aunt Olga. How often does Olga see her nephew?Discuss the corresponding problem involving the train to
(a) Let \(m, n\) be coprime integers, and suppose \(a\) is an integer which is divisible by both \(m\) and \(n\). Prove that \(m n\) divides \(a\).(b) Show that the conclusion of part (a) is false if \(m\) and \(n\) are not coprime (i.e., show that if \(m\) and \(n\) are not coprime, there exists
Jim plays chess every 3 days, and his friend Marmaduke eats spaghetti every 4 days. One Sunday it happens that Jim plays chess and Marmaduke eats spaghetti. How long will it be before this again happens on a Sunday?
Prove by induction that \(\Sigma_{r=1}^{n} r^{2}=\frac{1}{6} n(n+1)(2 n+1)\).Deduce formulae for\(1 \cdot 1+2 \cdot 3+3 \cdot 5+4 \cdot 7+\cdots+n(2 n-1)\) and \(1^{2}+3^{2}+5^{2}+\cdots(2 n-1)^{2}\).
(a) Work out \(1,1+8,1+8+27\) and \(1+8+27+64\). Guess a formula for \(\sum_{r=1}^{n} r^{3}\) and prove it.(b) Check that \(1=0+1,2+3+4=1+8\) and \(5+6+\cdots+9=8+27\). Find a general formula for which these are the first three cases. Prove your formula is correct.
Prove the following statements by induction:(a) For all integers \(n \geq 0\), the number \(5^{2 n}-3^{n}\) is a multiple of 11 .(b) For any integer \(n \geq 1\), the integer \(2^{4 n-1}\) ends with an 8 .(c) The sum of the cubes of three consecutive positive integers is always a multiple of 9 .(d)
Prove the following facts about complex numbers:(a) \(u+v=v+u\) for all \(u, v \in \mathbb{C}\).(b) \(u v=v u\) for all \(u, v \in \mathbb{C}\).(c) \((u+v)+w=(u+v)+w\) for all \(u, v, w \in \mathbb{C}\).(d) \(u(v+w)=u v+u w\) for all \(u, v, w \in \mathbb{C}\).(e) \(u(v w)=(u v) w\) for all \(u, v,
Prove the following, for all \(u, v \in \mathbb{C}\) :(a) \(\overline{u+v}=\bar{u}+\bar{v}\).(b) \(\overline{u v}=\bar{u} \bar{v}\).(c) \(|u|^{2}=u \bar{u}\).(d) \(|u v|=|u||v|\).
(a) Find the real and imaginary parts of \((\sqrt{3}-i)^{10}\) and \((\sqrt{3}-i)^{-7}\). For which values of \(n\) is \((\sqrt{3}-i)^{n}\) real?(b) What is \(\sqrt{i}\) ?(c) Find all the tenth roots of \(i\). Which one is nearest to \(i\) in the Argand diagram?(d) Find the seven roots of the
Prove the "Triangle Inequality" for complex numbers: \(|u+v| \leq|u|+|v|\) for all \(u, v \in \mathbb{C}\).
Express \(\frac{1+i}{\sqrt{3}+i}\) in the form \(x+i y\), where \(x, y \in \mathbb{R}\). By writing each of \(1+i\) and \(\sqrt{3}+i\) in polar form, deduce that\[ \cos \frac{\pi}{12}=\frac{\sqrt{3}+1}{2 \sqrt{2}}, \quad \sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2 \sqrt{2}} . \]
(a) Show that \(x^{5}-1=(x-1)\left(x^{4}+x^{3}+x^{2}+x+1\right)\). Deduce that if \(\omega=e^{2 \pi i / 5}\) then \(\omega^{4}+\omega^{3}+\omega^{2}+\omega+1=0\).(b) Let \(\alpha=2 \cos \frac{2 \pi}{5}\) and \(\beta=2 \cos \frac{4 \pi}{5}\). Show that \(\alpha=\omega+\omega^{4}\) and
Find a formula for \(\cos 4 \theta\) in terms of \(\cos \theta\). Hence write down a quartic equation (i.e., an equation of degree 4) that has \(\cos \frac{\pi}{12}\) as a root. What are the other roots of your equation?
Find all complex numbers \(z\) such that \(|z|=|\sqrt{2}+z|=1\). Prove that each of these satisfies \(z^{8}=1\).
Prove that there is no complex number \(z\) such that \(|z|=|z+i \sqrt{5}|=1\).
Show that if \(w\) is an \(n^{t h}\) root of unity, then \(\bar{w}=\frac{1}{w}\). Deduce that\[ \overline{(1-w)}^{n}=(w-1)^{n} . \]Hence show that \((1-w)^{2 n}\) is real.
Let \(n\) be a positive integer, and let \(z \in \mathbb{C}\) satisfy the equation\[ (z-1)^{n}+(z+1)^{n}=0 \](a) Show that \(z=\frac{1+w}{1-w}\) for some \(w \in \mathbb{C}\) such that \(w^{n}=-1\).(b) Show that \(w \bar{w}=1\).(c) Deduce that \(z\) lies on the imaginary axis.
Critic Ivor Smallbrain is discussing the film Sets, Lines and Videotape with his two chief editors, Sir Giles Tantrum and Lord Overthetop. They are sitting at a circular table of radius 1 . Ivor is bored and notices in a daydream that he can draw real and imaginary axes, with origin at the center
(a) Find the (complex) roots of the quadratic equation \(x^{2}-5 x+7-i=0\).(b) Find the roots of the quartic equation \(x^{4}+x^{2}+1=0\).(c) Find the roots of the equation \(2 x^{4}-4 x^{3}+3 x^{2}+2 x-2=0\), given that one of them is \(1+i\).
Solve \(x^{3}-15 x-4=0\) using the method for solving cubics.Now cleverly spot an integer root. Deduce that\[ \cos \left(\frac{1}{3} \tan ^{-1}\left(\frac{11}{2}\right)\right)=\frac{2}{\sqrt{5}} \]
Factorize \(x^{5}+1\) as a product of real linear and quadratic polynomials.
(a) Factorize \(x^{2 n+1}-1\) as a product of real linear and quadratic polynomials.(b) Write \(x^{2 n}+x^{2 n-1}+\cdots+x+1\) as a product of real quadratic polynomials.(c) Let \(\omega=e^{2 \pi i / 2 n+1}\). Show that \(\sum \omega^{i+j}=0\), where the sum is over all \(i\) and \(j\) from 1 to
Restless during a showing of the ten-hour epic First Among Inequalities, critic Ivor Smallbrain thinks of a new type of number, which he modestly decides to call "Smallbrain numbers." He calls an \(n\)-digit positive integer a Smallbrain number if it is equal to the sum of the \(n^{\text {th }}\)
Using Rules 5.1, show that if \(x>0\) and \(y0\) then \(\frac{1}{a}
For which values of \(x\) is \(x^{2}+x+1 \geq \frac{x-1}{2 x-1}\) ?
For which values of \(x\) is \(-3 x^{2}+4 x>1\) ?
(a) Find the set of real numbers \(x eq 0\) such that \(2 x+\frac{1}{x}
Prove that \(|x y|=|x||y|\) for all real numbers \(x, y\).
Find the range of values of \(x\) such that(i) \(|x+5| \geq 1\).(ii) \(|x+5|>|x-2|\).(iii) \(|x+5|
Prove the following inequaltities for any positive real numbers \(x, y\) :(i) \(x y^{3} \leq \frac{1}{4} x^{4}+\frac{3}{4} y^{4}\)(ii) \(x y^{3}+x^{3} y \leq x^{4}+y^{4}\).
Prove that if \(x, y, z\) are real numbers such that \(x+y+z=0\), then \(x y+\) \(y z+z x \leq 0\).
Show that \((50)^{3 / 4}\left(\frac{5}{\sqrt{2}}\right)^{-1 / 2}=10\).
Simplify \(2^{1 / 2} 5^{1 / 2} 4^{-1 / 4} 20^{1 / 4} 5^{-1 / 4} \sqrt{10}\).
What is the square root of \(2^{1234}\) ?What is the real cube root of \(3^{\left(3^{333}\right)}\) ?
Find an integer \(n\) and a rational \(t\) such that \(n^{t}=2^{1 / 2} 3^{1 / 3}\).
Which is bigger: \(100^{10000}\) or \(10000^{100}\) ?Which is bigger: the cube root of 3 or the square root of 2 ? (No calculators allowed!)
Find all real solutions \(x\) of the equation \(x^{1 / 2}-(2-2 x)^{1 / 2}=1\).
Prove that if \(x, y>0\) then \(\frac{1}{2}(x+y) \geq \sqrt{x y}\). For which \(x, y\) does equality hold?
When we want to add three numbers, say \(a+b+c\), we don't bother inserting parentheses because \((a+b)+c=a+(b+c)\). But with powers, this is not true \(-\left(a^{b}\right)^{c}\) need not be equal to \(a^{\left(b^{c}\right)}\) - so we must be careful. Show that this really is a problem, by finding

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A Concise Introduction To Pure Mathematics 4th Edition Martin Liebeck - Solutions

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