Instructions: Check that your test consists of 25
problems. Put your name in the space provided above.
Answer each multiple choice question below. Each problem is worth 4 points with
the same scoring procedure
as used on the quizzes. A blank page is provided at the end of this test for
your work.
1. If the base 10 number is converted to
base 3, then the result will be
2. Which of the following is equal to ?
3. Little Tommy has done the column addition problem to
the right. However, there is a question mark where a
number should be . What number should he put where
the question mark appears?


4. The figure to the right illustrates which multiplication
problem below?


5. Little Tommy has set up a lattice multiplication problem
to the right. What number should he put where the question
mark appears?


6. A person wants to mentally compute the sum
29 + 33 + 32 + 16 + 71 + 34 + 18.
The person notices that compatible numbers can be used to perform the mental
computation by pairing
the numbers in a convenient way. One of the numbers, however, appearing in the
sum above is not part
of a compatible pair. Which number is it?
7. What is the value of
7 × 4 × 103 × 25 × 25 × 8 × 5 ?
This is meant as a mental arithmetic problem (but I don’t care how you do it).
8. Which of the following numbers is the largest? (Note that you can derive the
answer by estimating rather
than by doing precise arithmetic .)
9. To use the chip model to compute (that
is, negative 3 minus positive 2), a student does the steps
in the order indicated below. One of the steps needs to be clearly changed in
order to make the student’s
use of the chip model correct. Which step needs to be changed?
(a) The student places down 3 red chips.
(b) Then the student puts down two red chips and two blue chips.
(c) Then the student removes the two red chips.
(d) Then the student concludes that the answer is negative five.
10. A student is in the process of using the number line model to compute
(that is, negative 4 minus
negative 4). She has made the rather elegant drawing below showing her work.
What is the next step she should do? (Be careful!! She is
NOT at the very beginning of the problem.
She is somewhere in the middle of it.)
(a) She should flip the witch so that the witch is looking in the opposite
direction.
(b) She should turn the witch upside down so that it is pointing upward.
(c) She should move the witch forward four units.
(d) She should move the witch backward four units.
11. In this problem, a, b and d are positive integers.
Each of the following statements is true for some choices
of a, b and d. However, only one of them is also not true for some choices of a,
b and d. Which one is it?
(a) If d  a and d  b, then d  (a + b).
(b) If d  a and d  b, then d  (a − b).
(c) If d  a and d  b, then d  (a + b).
(d) If d  a and d  b, then d  (a + b).
12. In this problem, a and b are positive integers. Three of the statements
below mean the same thing.
Which one does not mean the same thing as the other three?
(a) The number b is divisible by the number a.
(b) The number b is a factor of the number a.
(c) The number b divides the number a.
(d) The number b is a divisor of the number a.
13. The ten digit number N = 7777777728 has eight 7’s in it (not nine 7’s like
on the quiz) and is clearly
divisible by both 1 and 2. What is the total number of positive integers 10 that
divide N?
14. Which number below divides 2711814 ?
15. If 21 divides n, then how many of the numbers
must also divide n?
16. What is the least number divisible by each natural number less than or equal
to 20?
17. Which of the following is true about the least prime
that divides 149?
(a) The least prime is 1.
(b) The least prime is bigger than 1 and less than 10.
(c) The least prime is bigger than 10 and less than 100.
(d) The least prime is 149 since 149 is a prime.
18. What is the value of GCD ?
19. What is the value of LCM ?
20. The factoring tree to the right is missing some
numbers. What number belongs in the top
rectangle where a question mark appears?


21. John is taking a test that asks for the value of the
product
4 · 23 · 37 · 119 · 331 · 953 · 2027.
He panics and then realizes that he is taking a multiple choice test and all he
really has to do is to
eliminate three wrong choices. The various choices he is given are indicated
below. He remembers his
divisibility tests and is able to quickly determine the correct choice. What is
the correct choice?
22. What is the greatest common divisor of 21693204 and 21693264 ?
23. Note that the factor trees for 360 and 11250 are given
below. Which of the following is true about
GCD(360, 11250)?
24. You may want to use the factor tree for 11250 above
for this problem. Which of the following is a divisor
of 11250?
25. If a and b are positive integers and GCD(a, b) = 12,
then which of the following must be true about
LCM(a,b)?
(a) LCM(a, b) = ab/12
(b) LCM(a, b) > ab/12
(c) LCM(a, b) < ab/12
(d) none of these must be true