Test: Exponents/Powers - GMAT MCQ

Statement (1): n = 2x + 1, where x is a positive integer.

If we substitute the given expression into 4ⁿ, we get 4^{(2x + 1)}. When we expand this expression, the units digit of 4^(2x) will always be 6 (since 4² = 16, 4⁴ = 256, 4⁶ = 4096, etc.), and multiplying 6 by 4 will give a units digit of 4. Therefore, regardless of the value of x, the units digit of 4^n will always be 4. Statement (1) alone is sufficient to determine the units digit of 4^n.

Statement (2): n = 2k – 1, where k is a positive integer.

If we substitute the given expression into 4ⁿ, we get 4^{(2k – 1)}. Similarly, when we expand this expression, the units digit of 4^(2k) will always be 6, and multiplying 6 by 4 will give a units digit of 4. Therefore, regardless of the value of k, the units digit of 4^n will always be 4. Statement (2) alone is also sufficient to determine the units digit of 4ⁿ.

Since both statements (1) and (2) alone are individually sufficient to determine the units digit of 4ⁿ, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement (1): x^w = 1/2

From this statement, we know that the product of x and w is equal to 1/2. However, we don't have any specific values for x or w, so we cannot determine their individual values or the value of their product xw. Statement (1) alone is not sufficient to answer the question.

Statement (2): w = -1

This statement tells us the value of w, which is -1. However, we still don't have any information about the value of x. Without knowing the value of x, we cannot determine the value of xw. Statement (2) alone is not sufficient to answer the question.

Since neither statement alone is sufficient to determine the value of xw, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): x = 4n + 2

We are given that x can be expressed as 4n + 2, where n is a positive integer.

Substituting this expression into 3x + 1, we have:
3(4n + 2) + 1 = 12n + 6 + 1 = 12n + 7

To determine if the remainder is 0 when 3x + 1 is divided by 10, we need to check if 12n + 7 is divisible by 10.

If we take various values of n, we can see that the value of 12n + 7 will not be divisible by 10. For example, when n = 1, 12n + 7 = 19, which is not divisible by 10.

Therefore, statement (1) alone is sufficient to determine that the remainder is not 0 when 3x + 1 is divided by 10.

Statement (2): x > 4

This statement tells us that x is greater than 4. However, it does not provide any information about the divisibility of 3x + 1 by 10. Without knowing the specific value of x, we cannot determine the remainder. Statement (2) alone is not sufficient to answer the question.

Since statement (1) alone is sufficient to answer the question, but statement (2) alone is not, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): k is a prime number

For any prime number greater than 2, the units digit of its multiples will always be 2, 3, 7, or 8. Since 76 multiplied by any prime number greater than 2 will always have a units digit of 2, statement (1) alone is sufficient to determine the units digit of 76k.

Statement (2): k is greater than 2

This statement provides no specific information about the value of k or its divisibility, and it does not help determine the units digit of 76k. Statement (2) alone is not sufficient to answer the question.

Since statement (1) alone is sufficient to determine the units digit of 76k, while statement (2) alone is not, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement (1): x ≥ 3

This statement tells us that x is greater than or equal to 3. Since x is a positive integer, the possible values for x could be 3, 4, 5, and so on. Regardless of the specific value of x, the hundreds digit of 30x will always be 0. Therefore, statement (1) alone is sufficient to determine the value of the hundreds digit.

Statement (2): x/3 is an integer

This statement tells us that x is divisible by 3, meaning x is a multiple of 3. Again, regardless of the specific value of x, the hundreds digit of 30x will always be 0. Therefore, statement (2) alone is sufficient to determine the value of the hundreds digit.

Since both statements individually provide sufficient information to determine the value of the hundreds digit, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

To determine if 100x + y is divisible by 9, we need the sum of the digits to be divisible by 9.

Let's analyze each statement:

Statement (1): x = 5

Knowing the value of x alone does not provide any information about the value of y or the sum of the digits. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): y = 98

Knowing the value of y alone does not provide any information about the value of x or the sum of the digits. Therefore, statement (2) alone is not sufficient to answer the question.

Since neither statement alone is sufficient to determine if 100x + y is divisible by 9, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Conclusion:

Each statement alone provides enough information to determine the value of x+y.

The correct answer is: 4. EACH statement ALONE is sufficient to answer the question asked.

To determine if 25% of xy is greater than x, we need to know the values of both x and y.

Let's evaluate each statement:

Statement (1): x is positive

Knowing that x is positive does not provide any information about the value of y or the relationship between xy and x. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): y is even

Knowing that y is even does not provide any information about the value of x or the relationship between xy and x. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still don't have enough information to determine if 25% of xy is greater than x. We don't know the specific values of x and y or their relationship.

Therefore, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

To find the remainder when 5x is divided by y, we need to know the values of both x and y.

Let's analyze each statement:

Statement (1): x = 3

Knowing the value of x alone does not provide any information about the value of y. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): y = 4

Knowing the value of y alone does not provide any information about the value of x. Therefore, statement (2) alone is not sufficient to answer the question.

However, if we consider both statements together, we can determine the remainder when 5x is divided by y. Since x = 3 and y = 4, we have 5x = 5 * 3 = 15. When 15 is divided by 4, the remainder is 3.

Therefore, the answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): n(n+1) = 6

We can solve this equation by considering the possible values for n. We know that the product of two consecutive integers is 6. The only possible values for n are 2 and -3. Therefore, statement (1) alone is sufficient to determine the value of n.

Statement (2): 2⁽²ⁿ⁾ = 16

We can simplify this equation to 4n = 16, which further simplifies to n = 4. Therefore, statement (2) alone is sufficient to determine the value of n.

Since both statements individually provide sufficient information to determine the value of n, the answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.