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Properties of Integers


Is X an integer if AB > 0?

1. X = 2A - 5B
2. A = B

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

If A and B are both integers then 2A and 5yB are integers and thus X is an integer.

If they are not integers, however, we have no guarantee that 2A - 5B will be an integer.

Statement 1 is insufficient. Statement 2 tells us that A = B. Because this tells us nothing about X, Statement 2 is insufficient.

If we take Statements 1 and 2 together, we still can't determine if X is an integer because we do not know the value of A. 21.

Correct Answer - Choice E


Which of the following must be even if A and B are integers and AB^2 + 3B is odd?

A. B
B. A
C. AB + 3
D. AB - 3
E. A + 3

Answer

The first step is to completely factor our given expression. In this case, since B is in each term, we can factor it out as follows: AB^2 + 3B = B (AB + 3).

Let's focus on this factored version of the expression.

Since AB + 3 must be odd, and we know that the sum of two numbers is odd only when one of the numbers is odd and the other is even, AB must be even (because 3 is odd).

The product of two integers is even only if at least one of the terms is even. Therefore, since AB is even, and B is odd, A must be even. So, the correct answer choice is B.

Correct Answer - Choice B


Q) If X and Y are consecutive negative integers, is X greater than Y?

1. X - 1 and Y + 1 are consecutive negative integers.
2. X is an odd integer.

A.Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B.Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C.BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D.EACH statement ALONE is sufficient.
E.Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Choice A: Statement 1 tells us that X - 1 and Y + 1 are consecutive negative integers. As with positive numbers, subtracting from a negative number will make that number smaller and adding to it will make it larger. Therefore X must be greater than Y and Statement 1 alone is sufficient to answer the question.

Statement 2 tells us that X is an odd integer. This tells us nothing about the relative value of Y. Therefore, Statement 2 alone is insufficient to answer the question.




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