© Scarborough
Math 365
Exam 2
Spring 2013
Scarborough
Spring 2013
Math 365-501
Exam II
1
NEATLY PRINT NAME: ______________________________________
STUDENT ID: __________________________
DATE: _________________________
"On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work."
________________________________
Signature of student
Academic Integrity Task Force, 2004
http://www.tamu.edu/aggiehonor/FinalTaskForceReport.pdf
My signature in this blank allows my instructor to pass back my graded exam in class or allows me to pick up
my graded exam in class on the day the exams are returned. If I do not sign the blank or if I am absent from
class on the day the exams are returned, I know I must show my Texas A&M student ID during my instructor’s
office hours to pick up my exam.
Signature of student ____________________________________________
Write all solutions in the space provided as full credit will not be given without complete, correct accompanying
work, even if the final answer is correct. Use techniques taught in class to solve; do not use brute force (do not
use “list by exhaustion” unless that is the only way to solve the problem). Fully simplify all your answers, and
give exact answers unless otherwise stated. Make sure that you indicate your answer clearly by circling your
response.
Divisibility Test for 7
To see if a number is divisible by 7, double the digit in the ones place, and then subtract it from the remaining
digits of the number. If you get an answer divisible by 7, the original number is divisible by 7. If you do not
know if the new number is divisible by 7, repeat the process.
Example: 2625
Double the units’ digit of 5 to get 10. Subtract 10 from 262 to get 252. If we are not sure if 252 is divisible by 7
or not, repeat the process. Double 2 to get 4. Subtract 4 from 25 to get 21. Since 21 is divisible by 7, then 2625
is divisible by 7.
© Scarborough
Spring 2013
Math 365-501
Exam II
2
(10 pts: 1 pt for each TF) On problems 1 through 10, circle either “True” or “False.”
1. True or False: 0 | 17
2. True or False: The four lowest perfect squares are 1, 4, 9, and 16.
3. True or False: Each composite number can be written as a product of primes, in one and only one way,
except for the order of the prime factors in the product.
4. True or False: If 3 | a and 9 | a, then 27 | a.
5. True or False: The additive inverse of  3 is 3.
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6. True or False:  507  has 190 positive divisors.
7. True or False: If a | d and b | d , then  a  b  | d .
8. True or False: The perfect squares have an odd number of positive factors.
9. True or False: If b | a, then b is a factor or divisor of a, and a is a multiple of b.
10. True or False: Any positive integer with two distinct divisors is prime.
#11 – 15: Short Answer – You do NOT need to show work.
11. (5 pts) Circle the numbers below that divide 1,932,612.
2
3
4
5
6
8
9
11
22
12. (3 pts) Give an example of a 3-digit composite odd number: __________________________

13. (3 pts) What justifies  6  10  3  1    2  10  7  1    6  10  3  1    2  10  7  1  ? __________________
14. (3 pts) What is the largest prime needed to check to see if 329 is prime? ________________________
15. (3 pts) What is the greatest three-digit number that has exactly 3 positive divisors? _________________
WORKOUT – SHOW WORK
16. (5 pts) Fully simplify  2 4  36  3  2  3  2  5 2  .
17. (3 pts) Using the charged field model, calculate the difference  3   4 .
© Scarborough
Spring 2013
Math 365-501
Exam II
18. (4 pts) Using the stair-step model, illustrate and compute the prime factorization of 4725.
19. (5 pts) Prime factor 1 4 7 9  1 4 7 8 .
20. (3 pts) Using the number line model, calculate the difference 5   3 .
21. (3 pts) Using the charged field model, calculate the product   3    2  .
22. (5 pts) Rewrite 2 x  4 without the absolute value symbol; fully simplify.
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© Scarborough
Spring 2013
Math 365-501
Exam II
23. (4 pts) Completely factor 24 x 2  6 y 2 .
24. (4 pts) Use the intersection of sets to find the greatest common divisors of 15 and 9.
25. (3 pts) Use the colored rod method to find the lcm (3, 4).
lcm (3, 4) = ____________________
26. (6 pts) Use the Euclidean algorithm and theorem to compute the lcm (111, 74).
lcm (111, 74) = ____________________
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© Scarborough
Spring 2013
Math 365-501
Exam II
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27. (6 pts) Find all the one-digit values for a and b such that the five-digit number 37 a 4 b is divisible by 9.
Put your answers in the table. Leave any unneeded boxes blank.
a
b
28. (6 pts) Use the rectangle method to illustrate all the positive divisors of 18.
29. (4 pts) Use the definition of division of integers to find the quotient 16   8 , if it exists. If it does not
exist, use the division algorithm.
© Scarborough
Spring 2013
Math 365-501
30. (4 pts) Prove   a    b   ab if a , b  Z .
31. (4 pts) Justify why if x  y , then  x   y .
32. (4 pts) Use the definition of less than to prove  7   4 .
Five-Point Bonus: Prove    a   a .
Exam II
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