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Math 170(J. Wu) Answers to Homeworks
Homework 1: page 24

18.
\begin{diagram}
135+137+139+\cdots+579\\ =(1+3+5+7+\cdots+579)-(1+3+5+7+\cdots+133)\\ =290^2-67^2=84100-4489=79611.\end{diagram}

page 28

25. (see the book).

page 40

12.
\begin{diagram}
30,030=3,003\times 10 =1,001\times 3\times 10\\ =7\times 143\tim...
 ...es 3\times 2\times 5\\ =2\times 3\times5\times 7\times
11\times 13.\end{diagram}

14.
\begin{diagram}
100,100=100,1\times 10^2=7\times 11\times 13\times 2^2\times 5^2\\ =2^2\times 5^2\times 7\times 11\times 13.\\ \end{diagram}

16.

\begin{displaymath}
1,000,000=10^6=2^6\times 5^6.\end{displaymath}

32.

\begin{displaymath}
D(1,000,000)=7\times 7=49.\end{displaymath}

Page 45

18.
\begin{diagram}
S(30,030)=(1+2)\times (1+3)\times (1+5)\times (1+7)\times (1+11)...
 ...es 6\times 8\times 12\times 14\\ =2^9\times 3^3\times 7\\ =96768\\ \end{diagram}

20. Solution 1:
\begin{diagram}
S(100,100)=(1+2+2^2)\times (1+5+5^2)\times (1+7)\times (1+11)\times (1+13)\\ =7\times 31\times 8\times 12\times 14\\ =291648.\\ \end{diagram}
Solution 2:
\begin{diagram}
S(100,100)=\frac{2^3-1}{2-1}\cdot
\frac{5^3-1}{5-1}\cdot\frac{7^...
 ...c{13^2-1}{13-1}
=7\times 31\times 8\times 12\times 14\\ =291648.\\ \end{diagram}

22.
\begin{diagram}
S(1,000,000)=\frac{2^7-1}{2-1}\cdot\frac{5^7-1}{5-1}\\ =127\times\frac{78124}{4}=127\times 19531\\ =2480437\\ \end{diagram}

29. 230-1 cents$=\$10,737,418.34$.

(the answer in the book is wrong).

Extra Credit:

page 41

42.

60

page 45

28.

\begin{displaymath}
\frac{n^{k+1}-n^a}{n-1}\end{displaymath}



 

Jie Wu
11/13/1997