Math T104 Exam 2a

Individual Part

%Math T104 Exam 2a %Given March 1999 % \hsize=6.2truein \tolerance=3000 \parindent=1.5cm \rm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcount\qnno %\headline={\rm T104 Exam 2a\hfil {\it Individual Part}} \def\qn {\noindent\strut\global\advance\qnno by1% \hbox to\parindent{\the\qnno.\hfil}\hangindent\parindent\ignorespaces} \def\qnn#1 {\noindent\strut \hbox to\parindent{\phantom{\the\qnno. }#1\hfil}\hangindent\parindent\ignorespaces} \def\qne#1 {\noindent\strut\global\advance\qnno by1% \hbox to\parindent{\the\qnno. #1\hfil }\hangindent\parindent\ignorespaces} \def\elem#1 #2 #3 #4 { $\vcenter{\halign{\hfil##$\,$&\hfil##\cr &${\tt #1}_{#4}$\cr $#2$&${\tt #3}_{#4}$\cr \noalign{\smallskip\hrule}}}$} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \qn Answer the following questions about the number $1\,200\,030\,042$ and state the rule you used in arriving at your answer. \medskip \qnn a. Is it divisible by $2$? \bigskip \qnn b. Is it divisible by $3$? \bigskip \qnn c. Is it divisible by $4$? \bigskip \qnn d. Is it divisible by $5$? \bigskip \qnn e. Is it divisible by $6$? \bigskip \qnn f. Is it divisible by $9$?

\qne a. Is $1$ a prime number? \bigskip \qnn b. What is the prime factorization of $234$? \bigskip \qnn c. What is the prime factorization of $21$? \bigskip \qnn d. What is the least common multiple of $234$ and $21$? \bigskip \qnn e. What is the greatest common divisor of $234$ and $21$?

\noindent Addition and multiplication tables for modulo 10 arithmetic are given below. \medskip \hbox to\hsize{\indent \vbox{\offinterlineskip \vbox{\halign{\strut\vbox to0.4cm{\vfil \hbox to0.6cm{\hfil\tt #\hfil}\vfil}\vrule &&\vbox to0.4cm{\vfil\hbox to0.6cm{\hfil\tt #\hfil}\vfil}\cr +&0&1&2&3&4&5&6&7&8&9\cr \noalign{\hrule} 0&0&1&2&3&4&5&6&7&8&9\cr 1&1&2&3&4&5&6&7&8&9&0\cr 2&2&3&4&5&6&7&8&9&0&1\cr 3&3&4&5&6&7&8&9&0&1&2\cr 4&4&5&6&7&8&9&0&1&2&3\cr 5&5&6&7&8&9&0&1&2&3&4\cr 6&6&7&8&9&0&1&2&3&4&5\cr 7&7&8&9&0&1&2&3&4&5&6\cr 8&8&9&0&1&2&3&4&5&6&7\cr 9&9&0&1&2&3&4&5&6&7&8\cr }} }\hfill}

\noindent and \medskip \hbox to\hsize{\indent \vbox{\offinterlineskip \vbox{\halign{\strut\vbox to0.4cm{\vfil \hbox to0.6cm{\hfil\tt #\hfil}\vfil}\vrule &&\vbox to0.4cm{\vfil\hbox to0.6cm{\hfil\tt #\hfil}\vfil}\cr $\times$&0&1&2&3&4&5&6&7&8&9\cr \noalign{\hrule} 0&0&0&0&0&0&0&0&0&0&0\cr 1&0&1&2&3&4&5&6&7&8&9\cr 2&0&2&4&6&8&0&2&4&6&8\cr 3&0&3&6&9&2&5&8&1&4&7\cr 4&0&4&8&2&6&0&4&8&2&6\cr 5&0&5&0&5&0&5&0&5&0&5\cr 6&0&6&2&8&4&0&6&2&8&4\cr 7&0&7&4&1&8&5&2&9&6&3\cr 8&0&8&6&4&2&0&8&6&4&2\cr 9&0&9&8&7&6&5&4&3&2&1\cr }} }\hfill}

\qne a. Calculate ${\tt 3}\times{\tt 9}+{\tt 5}$ mod $10$. \bigskip \qnn b. Calculate ${\tt 3}\times({\tt 9}+{\tt 5})$ mod $10$. \bigskip \qne a. What is the identity for mod $10$ addition? \bigskip \qnn b. What is the identity for mod $10$ multiplication? \bigskip \qnn c. Which elements have inverses under mod $10$ multiplication?

\qne a. Write the fraction $1/2$ as a decimal. \bigskip \qnn b. Write the decimal $.34$ as a fraction in lowest terms. \bigskip \qnn c. Write the fraction $2/7$ as a decimal. \bigskip \qnn d. Write the decimal $.34\overline{6}$ as a fraction in lowest terms.

Group Part

%\headline={\rm T104 Exam 1a\hfil {\it Group Part}} \qnno=0 \qne a. Compute $\displaystyle {1\over 3}+{2\over 5}$. \bigskip \qnn b. Compute $\displaystyle {1\over 3}\times{2\over 5}$. \bigskip \qnn c. Explain why the method used for additing fractions is different than the method used for multiplying them. \bigskip \qnn d. Are there any fractions $\displaystyle {a\over b}$ and $\displaystyle {c\over d}$ such that $${a\over b}+{c\over d}={a+c\over b+d}\,\hbox{?}$$

\qn The following appears on pages 190--191 of the text. Finish the proof as indicated.

{\parindent=0pt A rational number is any number that can be expressed in the form $a/b$ where $a$ and $b$ are integers and $b$ is not equal to zero. To prove that a number is irrational, we must prove that it cannot be expressed as a ratio of integers. This means not only that we cannot do it, but in fact that no one can do it. To prove that something cannot be done, proof by contradiction is often used. We assume that it can be done, and in so doing arrive at a conclusion that is obviously false. Since a true premise cannot, by correct logical reasoning, result in a false conclusion, we can assume that our premise must have been false.

We will use proof by contradiction to show that the square root of two is not rational.

Assume that the square root of two is rational. Then we must be able to express $\sqrt 2$ in reduced fraction form. Thus there must be integers $a$ and $b$ with $b\ne 0$ such that $$\sqrt 2={a\over b}.$$ Furthermore, since we said that the fraction is in reduced form, we know that $a$ and $b$ have no common factors.

Beginning with the equation $$\sqrt 2={a\over b}$$ square both sides to get $$ 2={a^2\over b^2}\qquad\hbox{or}\qquad 2b^2=a^2.$$ To finish the proof, you must show that this statement leads to a contradiction.

Hint: the contradiction has to do with the fact that we required $a$ and $b$ to have no common factors. You will need to recall some of the ideas from number theory, for example, questions of odd and even and divisibility.