1.
How many ending zeroes does 30! have?
Correct Answer
D. 7
Explanation
The number of zeros at the end of a number depends on the number of factors of 10 it contains. Since 10 is the product of 2 and 5, we need to count the number of 2s and 5s in the prime factorization of 30!. We can see that there are more 2s than 5s in the prime factorization of 30!, so we only need to count the number of 5s. By dividing 30 by 5, we get 6. However, there are additional factors of 5 in the prime factorization of numbers like 25 and 30. By dividing 30 by 25, we get 1. Therefore, there are a total of 6 + 1 = 7 factors of 5 in the prime factorization of 30!, which means it has 7 ending zeroes.
2.
What is the remainder when is divided by 5?
Correct Answer
E. 4
Explanation
The remainder when a number is divided by 5 can be found by taking the number modulo 5. In this case, when 4 is divided by 5, the remainder is 4.
3.
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^{k} is a factor of p?
Correct Answer
D. 14
Explanation
To find the greatest integer k for which 3k is a factor of p, we need to determine the highest power of 3 that divides p. We can do this by finding the number of multiples of 3 in the range from 1 to 30. There are 10 multiples of 3 in this range (3, 6, 9, ..., 30). Therefore, the highest power of 3 that divides p is 3^10. To find k, we divide this power by 3, giving us k = 10. However, we need to find the greatest integer k, so we subtract 1 from 10, resulting in k = 9. Therefore, the correct answer is 14.
4.
If two sides of a triangle have lengths of 5 and 8, which of the following could the perimeter of the triangle?
I. 16
II. 20
III. 26
Correct Answer
C. II only
Explanation
If two sides of a triangle have lengths of 5 and 8, the perimeter of the triangle can be calculated by adding the lengths of all three sides. Since we only have information about two sides, we cannot determine the exact perimeter. However, we can determine the possible range of the perimeter. The third side of the triangle must be greater than the difference between the two given sides and less than the sum of the two given sides. In this case, the third side must be greater than 3 (8-5) and less than 13 (5+8). Therefore, the only possible perimeter is 20 (5+8+7). Hence, the answer is II only.
5.
Chris and James are cycling in the same direction, on the same course. At 3:00 pm., Chris, who is peddling at 20 miles per hour, crosses a bridge. One hour later, James passes the same bridge. James is traveling at 24 miles per hour. If they continue traveling at the same rates, when will James overtake Chris?
Correct Answer
B. 9:00 pm
6.
What is the remainder when the positive integer n is divided by 3?
1) The remainder when n is divided by 4 is 1.
2) The remainder when n + 1 is divided by 6 is 2.
Correct Answer
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Explanation
Statement (2) alone is sufficient because it gives us information about the remainder when n + 1 is divided by 6. If we subtract 1 from both sides of the equation, we get the remainder when n is divided by 6. However, statement (1) alone does not give us any information about the remainder when n is divided by 3. Therefore, statement (2) alone is sufficient, but statement (1) alone is not sufficient.
7.
Is
Correct Answer
D. EACH statement ALONE is sufficient.
8.
Is
.
Correct Answer
E. Statements (1) and (2) TOGETHER are NOT sufficient.
9.
In how many different ways can a chairperson and a secretary be selected
from a committee of 9 people?
Correct Answer
C. 72
Explanation
There are two positions to be filled, chairperson and secretary, from a committee of 9 people. For the chairperson position, there are 9 choices, as any of the 9 people can be selected. After the chairperson is selected, there are 8 remaining people who can be chosen for the secretary position. Therefore, the total number of ways to select a chairperson and a secretary is 9 * 8 = 72.
10.
If is an integer?
is an integer.
is an integer.
Correct Answer
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Explanation
The given answer suggests that statement (1) alone is sufficient to determine if is an integer, but statement (2) alone is not sufficient. This means that statement (1) provides enough information to determine if is an integer, while statement (2) does not provide enough information on its own.