(c) Scarborough
Fall 2015
Math 365-505
Exam II
1
Math 365
NEATLY PRINT NAME: _______________________________
Exam 2
STUDENT ID: ______________________
Fall 2015
DATE: ____________________________
"On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work."
________________________________
Signature of student
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 ____________________________________________
You are authorized to use a pencil, eraser, and your own TAMU student ID; use of anything else is a
violation of the Aggie Honor Code.
Note: It is a violation of the Aggie Honor Code to continue writing on the exam after time is called and in
doing so will result in a zero for this exam and will be reported to the Aggie Honor Council.
ALL CELL PHONES MUST BE TURNED OFF AND PLACED AT THE FRONT OF THE ROOM! It is
academic dishonesty to have any electronic devices, including cell phones, on your person during this exam.
Having an electronic device on your person can result in a zero on this exam and an F* for this course.
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.
(c) Scarborough
Fall 2015
Math 365-505
Exam II
2
(10 pts: 1 pt for each TF) On problems 1 through 10, circle either “True” or “False.”
1.
2.
3.
4.
5.
6.
True or False: The smallest prime number is 1.
True or False: If 5 | a and 25 | a, then 125 | a.
True or False: If a  b is a unique integer, then b | a .
True or False: Only perfect squares have an odd number of positive factors.
True or False: The additive inverse of 2 is  2 .
4
True or False:  80  has 10 positive divisors.
7. True or False: If a | d and b | d , then  a  b  | d .
8. True or False: For all integers n , if x  y , then nx  ny .
9. True or False: 0 | 4
10. True or False: Any integer with exactly two distinct, positive divisors is a prime number.
#11 – 17: Short Answer – You do NOT need to show work.
11. (5 pts) Circle the numbers below that divide 648,516
2
3
4
5
6
8
9
11
18
22
12. (3 pts) Give an example of a 3-digit composite odd number: __________________________

13. (3 pts) What justifies  4   6   4  6  ? _________________________________________
14. (3 pts) What is the largest prime needed to check to see if 227 is prime? ________________________
15. (3 pts) What is the greatest two-digit number that has exactly 3 positive divisors? _________________
16. (4 pts) Using what you learned in the prime factorization method, what is the least common
denominator, LCD, of among all of the fractions:
1
x ( x  2)
,
2
x
2
,
1
6x
,
5
3( x  2)( x  2)
.
LCD: ____________________________
17. (3 pts) Give three applications of negative numbers.
________________________
_________________________
_________________________
(c) Scarborough
Fall 2015
Math 365-505
Exam II
WORKOUT – SHOW WORK
18. (5 pts) Fully simplify  3 4  45  3  5  5  4   2  6 2  .
19. (3 pts) Using the charged field model, calculate the difference 2   5 .
20. (5 pts) Using the stair-step model, illustrate and compute the prime factorization of 540.
540 = _____________________________
21. (5 pts) Prime factor 9 9 1 1  9 9 1 0 .
22. (3 pts) Using the number line model, calculate the difference  6   8 .
3
(c) Scarborough
Fall 2015
Math 365-505
Exam II
4
23. (3 pts) Using the charged field model, calculate the product   2    3  .
24. (4 pts) Completely factor 45 x 2  20 y 2 .
25. (4 pts) Use the intersection of sets to find the greatest common divisors of 21 and 14.
gcd (21, 14) = ________________________________
26. (3 pts) Use the colored rod method to find the lcm (3, 4).
lcm (3, 4) = ____________________
27. (3 pts) Use the definition of division of integers to find the quotient 24   4 , if it exists. If it does not
exist in the integers, use the division algorithm.
(c) Scarborough
Fall 2015
Math 365-505
Exam II
5
28. (6 pts) Use the Euclidean algorithm and theorem to compute the lcm (12, 9).
lcm (12, 9) = ____________________
29. (6 pts) Find all the one-digit values for a and b such that the five-digit number 2 a 45 b is divisible by 9.
Put your answers in the table. Leave any unneeded boxes blank.
a
b
30. (5 pts) Prove 0  n  0 where n is a non-zero integer.
(c) Scarborough
Fall 2015
Math 365-505
Exam II
6
31. (4 pts) Prove   a    b   ab if a , b  Z .
32. (4 pts) Use the definition of less than for integers to prove  5   2 .
33. (3 pts) Precisely define (using the appropriate piece-wise function) the absolute value of x. Then
describe in words what the absolute value of a number represents.
Five-Point Bonus: State the Fundamental Theorem of Arithmetic.