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.
Answer:
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
