DSQ: Understanding the properties of integers - GMAT Quantitative

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

Statement 1 tells you that and are of unlike sign, that and are of unlike sign, and that and are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , , and , there can be either exactly zero or two negative numbers; and that of , , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

From Statement 1 it follows that is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that is a multiple of both 5 and 7; since now has at least four factors (1,5, 7, and 35), is composite.

Let be the missing digit.

If we assume only Statement 1, then the absolute value alternating sum of the digits must be a multiple of 11. That is:

is a multiple of 11. The only positive value of that does this is , which makes .

If we assume only Statement 2, then the sum of the digits must be a multiple of 3. That is,

is a multiple of 3. This happens if , so Statement 2 only narrows the last digit down to three possibilities.

Let be the missing digit.

If we assume only Statement 1, then the absolute value alternating sum of the digits must be a multiple of 11. That is:

is a multiple of 11. The only positive value of that does this is , which makes .

If we assume only Statement 2, then the sum of the digits must be a multiple of 3. That is,

is a multiple of 3. This happens if , so Statement 2 only narrows the last digit down to three possibilities.

Let be the missing digit.

If we assume only Statement 1, then the absolute value alternating sum of the digits must be a multiple of 11. That is:

is a multiple of 11. The only positive value of that does this is , which makes .

If we assume only Statement 2, then the sum of the digits must be a multiple of 3. That is,

is a multiple of 3. This happens if , so Statement 2 only narrows the last digit down to three possibilities.

Let be the missing digit.

If we assume only Statement 1, then the absolute value alternating sum of the digits must be a multiple of 11. That is:

is a multiple of 11. The only positive value of that does this is , which makes .

If we assume only Statement 2, then the sum of the digits must be a multiple of 3. That is,

is a multiple of 3. This happens if , so Statement 2 only narrows the last digit down to three possibilities.