Test: Exponents/Powers - GMAT MCQ

Let's simplify the expression:
3(4 + 4x) + 9y = 12 + 12x + 9y = 12x + 9y + 12.

Now, let's analyze each statement:

Statement (1): x = 25.
If x = 25, then the expression becomes 12(25) + 9y + 12 = 300 + 9y + 12 = 312 + 9y. We still don't have enough information to determine the remainder because the value of y is unknown. Statement (1) alone is not sufficient to answer the question.

Statement (2): y = 1.
If y = 1, then the expression becomes 12x + 9(1) + 12 = 12x + 9 + 12 = 12x + 21. We still don't have enough information to determine the remainder because the value of x is unknown. Statement (2) alone is not sufficient to answer the question.

Since neither statement alone is sufficient to determine the remainder, the answer is option B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): x^y + y^x is even.
The sum of xy + yx can be simplified to 2xy, which is even. This statement implies that 2xy is even, but it doesn't provide any information about the value of x or y individually. Therefore, statement (1) alone is not sufficient to determine if x is even.

Statement (2): y^x + 4x is odd.
If y^x + 4x is odd, it means that the sum of yx and 4x is odd. This implies that both terms must have the same parity (either both even or both odd) since the sum of two terms with different parity would be even. Since 4x is always even (for any positive integer x), it means that yx must also be even. However, we still don't know the specific values of x and y, so we cannot determine if x is even or odd. Statement (2) alone is not sufficient to determine if x is even.

Combining both statements, we know that 2xy is even (from statement 1) and yx is even (implied by statement 2). This means that both xy and yx must be even. If both xy and yx are even, it implies that x and y are both even. Therefore, by considering both statements together, we can conclude that x is even. Hence, both statements together are sufficient to answer the question.

The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): The unit digit of x is 2.
If the unit digit of x is 2, it means that x can be expressed in the form of 10a + 2, where a is an integer. However, knowing only the unit digit is not sufficient to determine if the square root of x is an integer. For example, if x is 12, its square root is not an integer. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): x is divisible by 3.
If x is divisible by 3, it means that x can be expressed in the form of 3b, where b is an integer. However, this statement also does not provide enough information to determine if the square root of x is an integer. For example, if x is 9, its square root is an integer, but if x is 6, its square root is not an integer. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have the information that x is a positive integer with a unit digit of 2 and is divisible by 3. In other words, x is a multiple of both 2 and 3. The only positive integer that satisfies this condition is 6, and its square root is an integer (2). Therefore, both statements together are sufficient to answer the question.

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

The properties of exponents state that the value of any number raised to the power of zero is one.

(1) x^w = 1 [Insufficient] w could be 1 or 0, so x could be any integer.

(2) w is even [Insufficient] By itself, this tells us nothing about x.

Both statements together reveal that w must be zero, but x could still be any integer.
The correct answer is E.

Statement (1): x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
If x is the smallest integer divisible by all integers from 21 to 24, it means that x is a multiple of their least common multiple (LCM). The LCM of 21, 22, 23, and 24 is 24. Therefore, x must be a multiple of 24. Let's consider a few values of x:

If x = 24, then x⁵ + 1 = 24⁵ + 1 = 796,594,177, which is an odd number.
If x = 48, then x⁵ + 1 = 48⁵ + 1 = 65,611,007,681, which is an odd number.
Based on the above examples, it seems that x⁵ + 1 is always an odd number when x is a multiple of 24. However, we cannot be certain that it holds true for all positive integers. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): 5x is an odd number.
If 5x is an odd number, it means that x must be an odd number as well since an even number multiplied by an odd number is always even. However, knowing that x is odd does not provide enough information to determine if x⁵ + 1 is an odd number. For example, if x = 3, then x⁵ + 1 = 3⁵ + 1 = 244, which is an even number. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements, we know that x is a positive integer that is divisible by all integers from 21 to 24, and 5x is an odd number. Since x must be a multiple of 24, we can substitute x = 24k, where k is a positive integer, into the expression x⁵ + 1:

x⁵ + 1 = (24k)⁵ + 1 = 24,000,000k⁵ + 1 = 2(12,000,000k⁵) + 1.

The expression 2(12,000,000k⁵) is always even, and adding 1 to an even number results in an odd number. Therefore, x⁵ + 1 is always an odd number when x is a multiple of 24.

Since statement (1) alone is sufficient to determine that x is a multiple of 24, and we have concluded that x⁵ + 1 is always an odd number when x is a multiple of 24, the answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): x⁴ is even, where x is a prime number. If x is a prime number, it means that x is greater than 1 and can only be divided evenly by 1 and itself. Considering that x⁴ is even, it implies that x⁴ must be divisible by 2. The only way for x⁴ to be divisible by 2 is if x is an even number. However, we do not know the specific value of x or any information about y from this statement. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): xy is odd. If xy is odd, it means that one of the variables x or y must be odd while the other is even. However, we do not know which variable is odd or even, and we do not have any information about their specific values. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements, we can deduce the following:

From statement (1), we know that x is a prime number and x⁴ is even. The only prime number that satisfies this condition is 2. Therefore, x = 2.

From statement (2), we know that xy is odd. Since x = 2, we have 2y is odd, which means y must be odd as well.

Now, with x = 2 and y being odd, we can determine the value of x + y, which is 2 + y. However, we still do not know the specific value of y. Therefore, even with both statements together, we cannot determine the exact value of x + y.

The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): 30n > 1000.
If 30n is greater than 1000, it means that the product of 30 and n is greater than 1000. We can rewrite this inequality as 30n > 1000. Dividing both sides by 30, we have n > 33.33... Since n is a positive integer, the smallest possible value of n that satisfies this inequality is 34. When n is 34, the value of 30n is 1020, which has a hundreds digit of 0. However, for any value of n greater than 34, the hundreds digit of 30n will always be 1. Therefore, statement (1) alone is sufficient to determine the value of the hundreds digit.

Statement (2): n is a multiple of 3.
If n is a multiple of 3, it means that n can be expressed as 3k, where k is an integer. Substituting this into 30n, we have 30(3k) = 90k. From this expression, we can see that the hundreds digit of 30n will always be 0 since the hundreds place will be occupied by a multiple of 10 (k), which does not affect the hundreds digit. Therefore, statement (2) alone is sufficient to determine the value of the hundreds digit.

Since each statement alone is sufficient to determine the value of the hundreds digit, the answer is option D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): x = 12.
If x = 12, then the product becomes (5 * 12)(2y)(3z) = 60(2y)(3z). The prime factorization of 60 is 2^2 * 3 * 5. Since we have a 2 and a 5 in the product, we can create trailing zeros by multiplying any number of pairs of 2s and 5s. However, statement (1) alone does not provide any information about y or z, so we cannot determine the number of trailing zeros.

Statement (2): x < y.
Statement (2) provides a relationship between x and y but does not give any specific values. Therefore, statement (2) alone does not provide enough information to determine the number of trailing zeros.

Combining both statements, we know that x = 12 and x < y. However, we still do not have any information about z. Without knowing the values of y and z, we cannot determine the number of trailing zeros.

Therefore, both statements together are sufficient to determine that x = 12 and x < y, but they are not sufficient to answer the question about the number of trailing zeros. The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): a = 1.
If a = 1, then the expression becomes 42(1) + 1 + b = 43 + b. Without knowing the value of b, we cannot determine the remainder when 43 + b is divided by 10. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): b = 2.
If b = 2, then the expression becomes 42a + 1 + 2 = 42a + 3. We still don't know the value of a, so we cannot determine the remainder when 42a + 3 is divided by 10. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements, we know that a = 1 and b = 2. Substituting these values into the expression 42a + 1 + b, we have 42(1) + 1 + 2 = 43. The remainder when 43 is divided by 10 is 3.

Therefore, statement (2) alone is sufficient to determine the remainder when 42a + 1 + b is divided by 10. The answer is option B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): 6a = 24 * 3b.
Dividing both sides of the equation by 6, we get a = 4b. From this equation, we can see that the value of a is dependent on the value of b. However, we do not have any information about the specific values of a or b, so we cannot determine the value of b uniquely. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): a + b = 5.
This statement provides a relationship between a and b, but it does not provide enough information to determine the specific value of b. Without additional information, we cannot uniquely determine the value of b. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements, we know that 6a = 24 * 3b from statement (1), and a + b = 5 from statement (2). From statement (1), we can rewrite it as a = 4b. Substituting this into statement (2), we have 4b + b = 5, which simplifies to 5b = 5. Dividing both sides by 5, we find that b = 1.

Therefore, statement (1) alone is sufficient to determine the value of b. The answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.