1. When the positive integer [latex]n[latex] is divided by [latex]3[latex], the remainder is [latex]2[latex] and when [latex]n[latex] is divided by [latex]5[latex], the remainder is [latex]1[latex]. What is the least possible value of [latex]n[latex]?
2. [latex]n[latex] is a positive integer that is divisible by [latex]6[latex].
| Quantity A | Quantity B |
| The remainder when [latex]n[latex] is divided by [latex]12[latex] | The remainder when [latex]n[latex] is divided by [latex]18[latex] |
3.
| Quantity A | Quantity B |
| The least number divisible by [latex]2, 3, 4, 5[latex], and [latex]6[latex] | The least number that is a multiple of [latex]2, 3, 4, 5[latex] and [latex]6[latex] |
4. Which of the following could be the units digit of [latex]{ 57 }^{ n }[latex]where [latex]n[latex] is a positive integer?
Indicate all such digits.
(A) [latex]0[latex]
(B) [latex]1[latex]
(C) [latex]2[latex]
(D) [latex]3[latex]
(E) [latex]4[latex]
(F) [latex]5[latex]
(G) [latex]6[latex]
(H) [latex]7[latex]
(I) [latex]8[latex]
(J) [latex]9[latex]
5.
| Quantity A | Quantity B |
| The last digit in the number [latex]{ 252 }^{ 56 }[latex] | The last digit in the number [latex]{ 152 }^{ 56 }[latex] |
6. The last digit of the positive even number [latex]n[latex] equals the last digit of [latex]{ n }^{ 2 }[latex]. Which one of the following could be [latex]n[latex] ?
(A) [latex]12[latex]
(B) [latex]14[latex]
(C) [latex]15[latex]
(D) [latex]16[latex]
(E) [latex]17[latex]
7. The number [latex]3[latex] divides [latex]a[latex] with a result of [latex]b[latex] and a remainder of [latex]2[latex]. The number [latex]3[latex] divides [latex]b[latex] with a result of [latex]2[latex] and a remainder of [latex]1[latex]. What is the value of [latex]a[latex] ?
(A) [latex]13[latex]
(B) [latex]17[latex]
(C) [latex]21[latex]
(D) [latex]23[latex]
(E) [latex]27[latex]
8. The remainder when the positive integer [latex]m[latex] is divided by [latex]7[latex] is [latex]x[latex]. The remainder when [latex]m[latex] is divided by [latex]14[latex] is [latex]x + 7[latex]. Which one of the following could [latex]m[latex] equal?
(A) [latex]45[latex]
(B) [latex]53[latex]
(C) [latex]72[latex]
(D) [latex]85[latex]
(E) [latex]100[latex]
9. Each of the two positive integers [latex]a[latex] and [latex]b[latex] ends with the digit [latex]2[latex]. With which one of the following numbers does [latex]a-b[latex] end?
(A) [latex]0[latex]
(B) [latex]1[latex]
(C) [latex]2[latex]
(D) [latex]3[latex]
(E) [latex]4[latex]
10. If each of the three nonzero numbers [latex]a, b[latex], and [latex]c[latex] is divisible by [latex]3[latex], then [latex]abc[latex] must be divisible by which one of the following the numbers?
(A) [latex]8[latex]
(B) [latex]27[latex]
(C) [latex]81[latex]
(D) [latex]121[latex]
(E) [latex]159[latex]
11. If [latex]n[latex] is a positive integer, which one of the following numbers must have a remainder of [latex]3[latex] when divided by any of the numbers [latex]4, 5[latex] and [latex]6[latex]?
(A) [latex]12n + 3[latex]
(B) [latex]24n + 3[latex]
(C) [latex]80n + 3[latex]
(D) [latex]90n + 2[latex]
(E) [latex]120n + 3[latex]
12. [latex]x[latex] and [latex]h[latex] are both positive integers. When x is divided by [latex]7[latex], the quotient is [latex]h[latex] with a remainder of [latex]3[latex]. Which of the following could be the value of [latex]x[latex]?
(A) [latex]7[latex]
(B) [latex]21[latex]
(C) [latex]50[latex]
(D) [latex]52[latex]
(E) [latex]57[latex]
13. [latex]616[latex] divided by [latex]6[latex] yields remainder [latex]p[latex], and [latex]525[latex] divided by [latex]11[latex] yields remainder [latex]q[latex]. What is [latex]p + q[latex]?
| Quantity A | Quantity B |
| [latex]x[latex] | [latex]3[latex] |
14. If [latex]p[latex] is divisible by [latex]7[latex] and [latex]q[latex] is divisible by [latex]6[latex], [latex]pq[latex] must have at least how many factors greater than [latex]1[latex]?
(A) [latex]1[latex]
(B) [latex]3[latex]
(C) [latex]6[latex]
(D) [latex]7[latex]
(E) [latex]8[latex]
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 [latex]463[latex]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
