Question

If \(\frac{61^{2}-1}{h}\) is an integer, then \(h\) could be divisible by each of the following EXCEPT: a. 8 b. 12 c. 15 d. 18 e. 31

Short answer

18

Step-by-step solution

- Simplify the Expression

Simplify the expression \ \( \frac{61^2 - 1}{h} \) as follows: \ \( 61^2 - 1 = (61-1)(61+1) = 60 \cdot 62 \).

- Factor the Product

Factorize each component: \ \( 60 = 2^2 \cdot 3 \cdot 5 \) and \ \( 62 = 2 \cdot 31 \) so that the product becomes \ \( 60 \cdot 62 = 2^3 \cdot 3 \cdot 5 \cdot 31 \).

- Analyze the Divisibility

Determine the divisibility of \ \( 60 \cdot 62 \) for each option: \ a) 8: Factor is 2^3, which is present. Pass. \ b) 12: Factor is 2^2 \cdot 3, which is present. Pass. \ c) 15: Factor is 3 \cdot 5, which is present. Pass. \ d) 18: Factor is 2 \cdot 3^2, one of the 3's is missing. Fail. \ e) 31: Factor is 31, which is present. Pass.

Key concepts

Divisibility Rules

Understanding divisibility rules is essential for cracking math problems quickly and efficiently. Divisibility rules help us determine whether one number can be divided by another without performing long division. For example, a number is divisible by 2 if its last digit is even. A number is divisible by 3 if the sum of its digits is divisible by 3. These rules simplify complex problems, making them easier to solve. In our exercise, the use of divisibility rules for numbers such as 8, 12, 15, 18, and 31 saves significant time by allowing us to quickly plug in factors and see if they evenly divide our given expression.

Factoring Numbers

Factoring is the process of breaking down a number into its prime components. This is crucial for understanding the fundamental building blocks of numbers and for solving a variety of math problems, including those on the GMAT. For instance, in the step-by-step solution provided, the number 60 is factored into \(2^2 \times 3 \times 5\) and 62 into \(2 \times 31\). This not only simplifies the expression but also makes it easier to analyze for divisibility. Understanding how to factor numbers quickly and accurately can greatly enhance problem-solving capabilities.

Integer Properties

Integers have specific properties that can be very useful in solving math problems. They can be positive, negative, or zero but do not include fractions or decimals. Knowing the properties of integers helps in simplifying expressions and solving equations. For example, when dealing with the expression \(61^2 - 1\), simplifying it to \(60 \times 62\) and then factoring both components take advantage of integer properties. Integers are closed under addition, subtraction, multiplication, and division (except when dividing by zero), which makes them predictable and easier to manipulate in mathematical problems.

GMAT Preparation

Preparing for the GMAT requires a deep understanding of mathematical principles, including divisibility rules, factoring, and integer properties. A systematic approach to solving problems, as shown in the given exercise, is crucial. Breaking down complex expressions, like \(61^2 - 1\), into simpler factors and then examining their divisibility can make even the toughest problems manageable. Effective GMAT preparation involves practicing these types of problems to become faster and more accurate. Besides practicing problems, understanding the fundamental concepts and honing problem-solving strategies can significantly improve performance on the test.