Test: Exponents/Powers - GMAT MCQ

To determine whether 3^x > 9^y, let's analyze the given statements:

Statement (1): x = 2y + 2

We can substitute this expression for x in the inequality:

3^{(2y + 2)} > 9^y

We know that 9 is equal to 3², so we can simplify further:

3^{(2y + 2)} > (3²)^y

3^{(2y + 2)} > 3^(2y)

Now we can compare the exponents:

2y + 2 > 2y

Since the y terms cancel out, we are left with 2 > 0, which is always true. Therefore, statement (1) alone is sufficient to conclude that 3^x > 9^y.

Statement (2): x = 3y

We can substitute this expression for x in the inequality:

3^(3y) > 9^y

Here, we have 3^3y > (3²)^y

3^3y > 3^2y

Now, comparing the exponents:

3y > 2y

Since y is positive, we can divide both sides by 2y without changing the inequality direction:

3/2 > 1

This inequality is also true. Therefore, statement (2) alone is also sufficient to conclude that 3^x > 9^y.

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

Statement (1): m < 0
This statement alone is not sufficient to determine whether m^n is an integer. For example, if m = -2 and n = 0.5, then m^n = (-2)^0.5 = √(-2), which is not an integer. However, if n is an integer, then m^n would be an integer regardless of the value of m. Therefore, statement (1) alone is not sufficient.

Statement (2): nm is an integer
This statement alone does not provide any information about the value of n or m individually. It only tells us that their product is an integer, but it doesn't tell us whether m^n is an integer. Therefore, statement (2) alone is not sufficient.

Since neither statement alone is sufficient to answer the question, we need additional information to determine whether m^n is an integer. Therefore, the answer is (E) Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

Statement (1): When the sum 23x + 25y is divided by 10, the remainder is 8.
We can rewrite this statement as (23x + 25y) ≡ 8 (mod 10). Simplifying further, we get (3x + 5y) ≡ 8 (mod 10).

From this information alone, we cannot directly determine if x + y is divisible by 4. For example, if x = 1 and y = 3, then (3x + 5y) ≡ (3 + 15) ≡ 8 (mod 10), but x + y is not divisible by 4. However, if x + y is divisible by 4, then (3x + 5y) ≡ 8 (mod 10) would hold true. Therefore, statement (1) alone is not sufficient.

Statement (2): When 22y is divided by 10, the remainder is 8.
We can rewrite this statement as 22y ≡ 8 (mod 10). Simplifying further, we get 2y ≡ 8 (mod 10).

Similar to statement (1), we cannot determine if x + y is divisible by 4 based solely on this information. For example, if y = 4, then 2y ≡ 8 (mod 10), but we don't know the value of x or if x + y is divisible by 4. Therefore, statement (2) alone is not sufficient.

However, by combining both statements, we have the system of congruences:
(3x + 5y) ≡ 8 (mod 10)
2y ≡ 8 (mod 10)

From the second congruence, we can deduce that y ≡ 4 (mod 10). This means that y must end with the digit 4. We can then substitute this result into the first congruence to obtain:
(3x + 5(10a + 4)) ≡ 8 (mod 10)
3x + 50a + 20 ≡ 8 (mod 10)
3x ≡ 8 - 20 (mod 10)
3x ≡ -12 (mod 10)
3x ≡ 8 (mod 10)

This congruence tells us that 3x ends with the digit 8. The only possible positive integer value for x that satisfies this condition is x = 6. When x = 6, the sum x + y is 6 + (10a + 4), which simplifies to 10a + 10, and is clearly divisible by 4.

Therefore, by considering both statements together, we can determine that the sum x + y is divisible by 4. Thus, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): y is odd
If y is odd, then the sum xa + yb will always be even. This is because when x is even, xa is always even, and since y is odd, yb is always odd. Adding an even number to an odd number always results in an even number. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2): a! = b!
The factorial of a number a! represents the product of all positive integers from 1 to a. This statement alone does not provide any information about the parity (evenness or oddness) of a or b. It only tells us that the factorials of a and b are equal. Without further information, we cannot determine the evenness or oddness of xa or yb based solely on this statement. Therefore, statement (2) alone is not sufficient.

Since statement (1) alone is sufficient to determine that the expression xa + yb is even when x is even, but statement (2) alone is not sufficient, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): a² has a units digit of 9.
This statement tells us that the square of a has a units digit of 9. From this information, we can deduce that the units digit of a must be 3 or 7, as the only two possible digits whose squares end in 9. However, this information alone is not sufficient to determine the units digit of a¹⁸, as we need to know the specific value of a. Therefore, statement (1) alone is not sufficient.

Statement (2): a⁷ has a units digit of 3.
This statement provides information about the units digit of a⁷, which is 3. From this information, we can deduce that the units digit of a must be 7, as the only digit whose seventh power ends in 3. However, similar to statement (1), this information alone is not sufficient to determine the units digit of a^18. Therefore, statement (2) alone is not sufficient.

By considering each statement separately, we can see that both statements alone are sufficient to determine the units digit of a¹⁸. In statement (1), we find that a can only be 3 or 7, and in statement (2), we find that a must be 7. Regardless of which value a takes (3 or 7), the units digit of a¹⁸ will always be 1. Therefore, each statement alone is sufficient to answer the question.

Thus, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement (1): m is a divisor of 16.
This statement tells us that m is a factor of 16, which means m can take on the values 1, 2, 4, 8, or 16. However, it does not provide any information about the value of n. Therefore, statement (1) alone is not sufficient to determine if m + n is odd.

Statement (2): The units digit of 12m + 28n is 6.
This statement gives us information about the units digit of the sum 12m + 28n. Since the units digit is 6, we know that the sum ends in 6. However, it does not provide any information about whether the sum is odd or even. For example, if m = 1 and n = 2, the sum 12m + 28n is 76, which is even. But if m = 1 and n = 1, the sum is 40, which is even. Therefore, statement (2) alone is not sufficient.

By considering both statements together, we have the information that m is a divisor of 16 and the units digit of 12m + 28n is 6. From statement (1), we know that m can take on the values 1, 2, 4, 8, or 16. When we consider the units digit of 12m + 28n, we can see that for all these values of m, the units digit of the sum will always be 6. However, this still does not determine whether the sum m + n is odd or even.

For example, if m = 2 and n = 4, then m + n = 6, which is even. But if m = 1 and n = 5, then m + n = 6, which is odd.

Therefore, when both statements are considered together, we still cannot determine whether m + n is odd or even. Thus, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): a is even, b is a multiple of four.
If a is even, then 2a is a multiple of 4. Additionally, if b is a multiple of four, then 5b is also a multiple of 4. Since both 2a and 5b are multiples of 4, the exponent 2a + 5b is a multiple of 4. As a result, 18^{(2a + 5b)} will end in the same digit as 18⁴, as the unit's digit repeats every 4 powers of 18.

The unit's digit of 18⁴ is 4. Therefore, statement (1) alone is sufficient to determine the digit at the unit's place.

Statement (2): b = 12
This statement gives the value of b as 12. However, it does not provide any information about a or the exponent 2a + 5b. Without knowing the value of a, we cannot determine the digit at the unit's place. Therefore, statement (2) alone is not sufficient.

Since statement (1) alone is sufficient to determine the digit at the unit's place, but statement (2) alone is not sufficient, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): an = 64
This statement tells us that the product of the first 8 positive integers is equal to 64. However, it does not provide any information about the value of n. Therefore, statement (1) alone is not sufficient to determine the value of a.

Statement (2): n = 6
This statement provides a specific value for n, which is 6. However, it does not give any information about the product of the first 8 positive integers or the value of a. Therefore, statement (2) alone is not sufficient to determine the value of a.

By considering both statements together, we have the information that n = 6 and an = 64. Substituting the value of n into the equation, we get a^6 = 64. Taking the sixth root of both sides, we find a = 2.

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

Statement (1): A is divisible by 210.
This statement alone does not provide direct information about the divisibility of C by 5. Although A is divisible by 210, which is a multiple of 5, it doesn't guarantee that the fraction A/B reduces to a form where C is divisible by 5. For example, if A = 420 and B = 840, the fraction A/B equals 1/2, and C is not divisible by 5. Therefore, statement (1) alone is not sufficient.

Statement (2): B = 7x, where x is a positive integer.
This statement alone does not provide any information about C. It only gives information about the form of B, which is a multiple of 7. This information is not enough to determine if C is divisible by 5. For example, if B = 14, C could be any positive integer, and its divisibility by 5 cannot be determined solely from statement (2). Therefore, statement (2) alone is not sufficient.

By considering both statements together, we can combine the information. From statement (1), we know that A is divisible by 210, and from statement (2), we know that B is a multiple of 7. Combining these statements, we have A = 210a and B = 7x, where a and x are positive integers.

If we substitute these values into the fraction A/B = C/D, we get (210a)/(7x) = C/D. Simplifying further, we have (30a)/x = C/D. From this equation, we can deduce that x must be a multiple of 30, since the numerator on the left side of the equation contains 30. Since x is a multiple of 30, it is divisible by both 5 and 6.

Therefore, when both statements are considered together, we can determine that C is divisible by 5. Thus, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): k is divisible by 2⁶.
If k is divisible by 2⁶, it means that k is a multiple of 2⁶. However, being divisible by 2⁶ does not guarantee that k is equal to 2r for some positive integer r. For example, if k = 2⁶, it is divisible by 2⁶, but it cannot be expressed as 2r since r would have to be 13, which is not a positive integer. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): k is not divisible by any odd integer greater than 1.
This statement tells us that k is not divisible by any odd prime number greater than 1. In other words, k does not have any odd prime factors. However, this statement also does not provide enough information to determine if k can be expressed as 2r for some positive integer r. For example, if k = 15, it is not divisible by any odd integer greater than 1, but it cannot be expressed as 2r since it does not have a factor of 2. Therefore, statement (2) alone is not sufficient to answer the question.

By evaluating both statements together, we can see that if k is divisible by 26 (statement 1) and does not have any odd prime factors (statement 2), then k must be divisible by 2 and 13. This implies that k is equal to 2 multiplied by a positive integer (r) since it has a factor of 2. Therefore, statement (2) alone is sufficient to answer the question.

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