Quantity A
The number of distinct positive
factors of 10
Quantity B
The number of distinct prime
factors of 210
4.
Quantity A
The least common multiple of 22
and 6
Quantity B
The greatest common factor of 66
and 99
5. The number of students who attend a school could be divided among 10,
12, or 16 buses, such that each bus transports an equal number of students.
What is the minimum number of students that could attend the school?
(A)
120
(B)
160
(C)
240
(D)
320
(E)
480
6.
Quantity A
The number of distinct prime
factors of 27
Quantity B
The number of distinct prime
factors of 18
7. How many factors greater than 1 do 120, 210, and 270 have in common?
(A)
One
(B)
Three
(C)
Six
(D)
Seven
(E)
Thirty
8. Company H distributed $4,000 and 180 pencils evenly among its
employees, with each employee getting an equal integer number of dollars
and an equal integer number of pencils. What is the greatest number of
employees that could work for company H?
(A)
9
(B)
10
(C)
20
(D)
40
(E)
180
9.
n
is divisible by 14 and 3. Which of the following statements must be true?
Indicate all such statements.
12 is a factor of
n
.
21 is a factor of
n
.
n
is a multiple of 42.
10. Positive integers
a
and
b
each have exactly four positive factors. If
a
is a
one-digit number and
b
=
a
+ 9, what is the value of
a
?
11. Ramon wants to cut a rectangular board into identical square pieces. If the
board is 18 inches by 30 inches, what is the least number of square pieces
he can cut without wasting any of the board?
(A) 4
(B) 6
(C) 9
(D)
12
(E)
15
12. When the positive integer
x
is divided by 6, the remainder is 4. Each of
the following could also be an integer EXCEPT
(A)
(B)
(C)
(D)
(E)
13. If
x
y
= 64 and
x
and
y
are positive integers, which of the following could
be the value of
x
+
y
?
Indicate all such values.
2
6
7
8
10
12
14. If
k
is a multiple of 24 but not a multiple of 16, which of the following
cannot be an integer?
(A)
(B)
(C)
(D)
(E)
15. If
a
= 16
b
and
b
is a prime number greater than 2, how many positive
distinct factors does
a
have?
16. If
a
and
b
are integers such that
a
>
b
> 1, which of the following cannot
be a multiple of either
a
or
b
?
(A)
a
– 1
(B)
b
+ 1
(C)
b
– 1
(D)
a
+
b
(E)
ab
17. 616 divided by 6 yields remainder
p
, and 525 divided by 11 yields
remainder
q
. What is
p
+
q
?
18. If
x
is divisible by 18 and
y
is divisible by 12, which of the following
statements must be true?
Indicate all such statements.
x
+
y
is divisible by 6.
xy
is divisible by 48.
is divisible by 6.
19. If
p
is divisible by 7 and
q
is divisible by 6,
pq
must have at least how
many factors greater than 1?
(A)
One
(B)
Three
(C)
Six
(D)
Seven
(E)
Eight
20. If
r
is divisible by 10 and
s
is divisible by 9,
rs
must have at least how
many positive factors?
(A)
Two
(B)
Four
(C)
Twelve
(D)
Fourteen
(E)
Sixteen
21. If
t
is divisible by 12, what is the least possible integer value of
a
for
which
might not be an integer?
(A)
2
(B)
3
(C)
4
(D)
5
(E)
6
22. If
a, b
, and
c
are multiples of 3 such that
a
>
b
>
c
> 0, which of the
following values must be divisible by 3?
Indicate all such values.
a
+
b
+
c
a
–
b
+
c
23. New cars leave a car factory in a repeating pattern of red, blue, black, and
gray cars. If the first car to exit the factory was red, what color is the 463rd
car to exit the factory?
(A)
Red
(B)
Blue
(C)
Black
(D)
Gray
(E)
It cannot be determined from the information given.
24. Jason deposits money at a bank on a Tuesday and returns to the bank 100
days later to withdraw the money. On what day of the week did Jason
withdraw the money from the bank?
(A)
Monday
(B)
Tuesday
(C)
Wednesday
(D)
Thursday
(E)
Friday
25.
x
and
h
are both positive integers. When
x
is divided by 7, the quotient is
h
with a remainder of 3. Which of the following could be the value of
x
?
(A)
7
(B)
21
(C)
50
(D)
52
(E)
57
26. When
x
is divided by 10, the quotient is
y
with a remainder of 4. If
x
and
y
are both positive integers, what is the remainder when
x
is divided by 5?
(A)
0
(B)
1
(C)
2
(D)
3
(E)
4
27. What is the remainder when 13
17
+ 17
13
is divided by 10?
28. If
n
is an integer and
n
3
is divisible by 24, what is the largest number that
must be a factor of
n
?
(A)
1
(B)
2
(C)
6
(D)
8
(E)
12
10! is divisible by 3
x
5
y
, where
x
and
y
are positive integers.
29.
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