Question

If integers \(a\) and \(b\) are distinct factors of \(30,\) which of the following CANNOT be a factor of \(30 ?\) I. \(a b+b^{2}\) II. \((a+b)^{2}\) III. \(a+b\) A. I only B. II only C. III only D. I and II only E. I, II, and III

Short answer

D. I and II only

Step-by-step solution

Identify factors of 30

First, identify all distinct factors of 30. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

Evaluate expression I

Consider expression I: \(ab + b^2\). Since both \(a\) and \(b\) are factors of 30, \(ab\) is also a multiple of 30. Adding \(b^2\) could result in \(ab + b^2\) being a factor or not a factor of 30. For instance, if \(a = 2\) and \(b = 5\), then \(ab + b^2 = 2 \times 5 + 5^2 = 10 + 25 = 35\), which is not a factor of 30. Therefore, I cannot be a factor of 30.

Evaluate expression II

Consider expression II: \((a+b)^2\). Since \(a\) and \(b\) are factors of 30, the sum \(a + b\) can vary. Squaring \(a + b\) does not necessarily guarantee that it will be a factor of 30. For instance, if \(a = 1\) and \(b = 5\), then \((a+b)^2 = (1+5)^2 = 6^2 = 36\), which is not a factor of 30. Therefore, II cannot be a factor of 30.

Evaluate expression III

Consider expression III: \(a + b\). Since \(a\) and \(b\) are factors of 30, the sum \(a + b\) can sometimes be a factor of 30. For example, if \(a = 1\) and \(b = 5\), then \(a + b = 1 + 5 = 6\), which is a factor of 30. Therefore, it is possible for III to be a factor of 30.

Determine which cannot be factors

Based on the evaluations, expressions I (\(ab + b^2\)) and II (\((a+b)^2\)) cannot be factors of 30. Hence, the correct answer is D.

Key concepts

Integer Factors

Understanding integer factors is key for solving GMAT math problems. Factors of a number are integers that divide the number without leaving a remainder. For example, the factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30. These are all the numbers you can multiply together to get 30. Knowing the factors of a number helps us determine how other expressions involving those factors behave. It is crucial to be proficient with identifying these factors, especially for exams like the GMAT.

GMAT Problem-Solving

When facing GMAT problem-solving questions, always start by understanding what is given and what is asked. Break down the problem as shown in the exercise by listing given information first. Here, the problem involves finding which of the listed expressions cannot be factors of 30. By breaking the problem into smaller steps, such as identifying the factors of 30 first, then evaluating each expression, we can systematically approach the solution. This methodical process is effective for tackling any GMAT math problem.

Mathematical Expressions

In our exercise, we deal with various mathematical expressions involving factors of 30. For instance, the first expression is \(ab + b^2\). By substituting different pairs of factors, we see if the results are also factors of 30. A clear understanding of how to manipulate and analyze these expressions is essential. The second expression \((a+b)^2\), involves combinatorial summing and squaring, which also needs evaluation with the known factors. Therefore, this exercise helps in grasping how mathematical expressions work with integer factors.

Exam Preparation

Preparing for exams like the GMAT requires not just understanding the content, but also practicing strategy. Solving similar problems, timing oneself, and reviewing solutions in a step-by-step manner, as shown in the exercise, are excellent ways to prepare. Focus on mastering concepts such as integer factors and mathematical expressions, as well as refining problem-solving techniques. Being well-prepared means you are familiar with common question types and can approach each problem with confidence.