1. When the positive integer  $n$ is divided by $3$ , the remainder is  $2$ and when  $n$ is divided by $5$ , the remainder is $1$ . What is the least possible value of $n$ ?

2.  $n$ is a positive integer that is divisible by $6$ .

 Quantity A Quantity B The remainder when  $n$ is divided by $12$ The remainder when  $n$ is divided by $18$

3.

 Quantity A Quantity B The least number divisible by $2, 3, 4, 5$ , and $6$ The least number that is a multiple of $2, 3, 4, 5$ and $6$

4.  Which of the following could be the units digit of ${ 57 }^{ n }$ where  $n$ is a positive integer?
Indicate all such digits. 

(A) $0$ 
(B) $1$
(C) $2$
(D) $3$
(E) $4$
(F) $5$ 
(G) $6$
(H) $7$
(I) $8$
(J) $9$

5.

 Quantity A Quantity B The last digit in the number ${ 252 }^{56 }$ The last digit in the number ${ 152 }^{56 }$

6.   The last digit of the positive even number $n$ equals the last digit of ${ n }^{ 2 }$ . Which one of the following could be  $n$ ?

(A) $12$
(B) $14$
(C) $15$
(D) $16$
(E) $17$

7.  The number $3$ divides $a$ with a result of $b$ and a remainder of $2$ . The number $3$ divides $b$ with a result of $2$ and a remainder of $1$ . What is the value of $a$ ?

(A) $13$
(B) $17$
(C) $21$
(D) $23$
(E) $27$

8. The remainder when the positive integer $m$ is divided by $7$ is $x$ . The remainder when $m$ is divided by $14$ is $x + 7$ . Which one of the following could $m$ equal?

(A) $45$
(B) $53$
(C) $72$
(D) $85$
(E) $100$

9.  Each of the two positive integers $a$ and $b$ ends with the digit $2$ . With which one of the following numbers does $a-b$ end?

(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $4$

10.   If each of the three nonzero numbers $a, b$ , and $c$ is divisible by $3$ , then $abc$ must be divisible by which one of the following the numbers?

(A) $8$
(B) $27$
(C) $81$
(D) $121$
(E) $159$

11.  If $n$ is a positive integer, which one of the following numbers must have a remainder of $3$ when divided by any of the numbers $4, 5$ and $6$ ?

(A) $12n + 3$
(B) $24n + 3$
(C) $80n + 3$
(D) $90n + 2$
(E) $120n + 3$

12.   $x$ 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) $57$

13. $616$ divided by $6$ yields remainder $p$ , and $525$ divided by $11$ yields remainder $q$ . What is $p + q$ ?

 Quantity A Quantity B $x$ $3$

14.  If $p$ is divisible by $7$ and $q$ is divisible by $6$ , $pq$ must have at least how many factors greater than $1$ ?

(A) $1$
(B) $3$
(C) $6$
(D) $7$
(E) $8$

15. 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 $463$ rd car to exit the factory?

(A) red
(B) blue
(C) black
(D) gray
(E) It cannot be determined from the information given.

1 11
2 D
3 C
4 B,D,H,J
5 C
6 A,C
7 D
8 D
9 B
10 A
11 E
12 D
13 12
14 D
15 C