MATH 70
Exam 1 Review
Fall 2013
Exam 1 will be on Monday, September 23 and will cover material on Chapter 2. The
following are sample problems similar to those that will be on 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 player 1
remove 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
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c) (16  24)mod13
d) 26 mod 30
8. If it is 10:00 now, what time will it be in 32 hours?
9. The UPC on a yogurt cup is 0 70470 00302 3. Is this a valid UPC number?
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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.
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14. Prove that 0.79999999…..=
0.8
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15. What characterizes the decimal expansion of a rational number?
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. 6:00
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 repeats without
terminating.
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