Math 70
Exam 1 Review
Fall 2009
Exam 1 will be on Friday, September 25 and will cover material on Chapter 2 in the
textbook and Chapter 1 in Zero: The Biography of a Dangerous Idea. The following are
sample problems similar to those that will be on the exam. You may use one 8 in by 11
in sheet (front and back) of notes for the exam. It must be a sheet that you have made
yourself, but can contain any information you like. The “note sheet” will be collected
with your exam on the day of the exam. I encourage you to bring a calculator to the
exam. You will not be allowed to use your cell phone as a calculator.
Sample Exam Questions
1. Each box of animal crackers contains two servings. Each serving consists of
exactly 12 crackers. There are exactly 18 different shapes of crackers. Are there
always two crackers of the same shape in each box? Explain why or why not.
2. If you play Fibonacci Nim with 14 sticks, how many sticks should be removed in
the first move using the winning strategy?
3. To what number does the ratio of consecutive Fibonacci numbers converge?
Name the number and express it numerically.
4. Let p stand for a prime number that is larger than 3. Can p+1 ever be a prime
number? Why or why not?
5. Are there infinitely many non-primes? Explain.
6. Define a  b mod n . Illustrate your definition with an example.
7. Evaluate the following using modular arithmetic:
a) 42 mod 5
b) 157 mod 3

c) (16  24)mod13
d) 26 mod 30
8. If it is 10:00 now, what time was it 32 hours ago?
9. The UPC on a yogurt cup is 0 70470 00302 3. Is this a valid UPC number?

10.
As a receiver of a message encoded using RSA encryption, using p  5 , q  11 ,
and r  3 , find s.
11. As a sender, use n  55 and r = 3 as above to encode the message “GO” using
RSA encryption.
5 3
12. Which of these are irrational? 12 ,
, 3.36
12
13. Express the decimal 3.373737373737…. as a fraction. Don’t worry about
reducing to lowest terms.


14. Prove that 0.79999999…..=
0.8
 expansion of a rational number?
15. What characterizes the decimal
16. Who were the first people to develop a need for zero? What was the need? Why
were the Greeks and other peoples so afraid of the number zero? Which peoples
were not scared of zero and used it as the number we do today?
Answers to Sample Exam Questions:

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1. There are always two crackers of the same shape in each box. There are only 18
shapes and 24 crackers in each box, so according to the Pigeonhole Principle,
each box must have at least 2 of one shaped cracker in each box.
2. 1 stick should be removed.
1 5
3.  
2
4. p + 1 can never be prime because it would be an even number.
5. Yes. You can take any natural number, m, and always create a non-prime larger
than it by many methods, for instance doubling it or tripling it. The larger number
you get will always be non-prime because it will be divisible by m and the
number you multiplied by m to get it. Since there are infinitely many natural
numbers, then you can do this infinitely and there will be infinitely many nonprime numbers.
6. a  b mod n means that b is the remainder of a divided by n. For example,
17  3mod 7 means that 3 is the remainder when you divide 17 by 7
( 17  (7  2)  3).
7. a) 2 b) 1 c) 7 d) 26
8. 2:00
9. Yes.
10. s = 7
11. GO is encrypted as 13=M, 20=T
12. 12
M  3.373737....
100M  337.373737....
13. 99M  334
334
M
99
14.
0.79999....  M
7.9999...  10M
79.9999...  100M
72  90M
72
M
90
0.8  M
15. The decimal expansion of a rational numbers either terminates or repeates without
terminating.
16. Babylonians developed zero to be a place holder. The Greeks and other peoples
were scared of zero because they believed that before the earth was created there
was a dark void and zero was representative of that dark void where no life lived.
The Mayans were not scared of the number zero and used it to represent the idea
of nothing.