Question

The function \(\&(\mathrm{x})\) is a clock function; it gives values between 0 and 12. \&(13) returns the value 1 , for example. What, then, is the result of \(\&(36)\) divided by \(\&(12) ?\) (A) 2 (B) 3 (C) 12 (D) 6 (E) 1

Short answer

After reassessment, \(\frac{\&(36)}{\&(12)}=\frac{0}{0}\) and division by zero is undefined.

Step-by-step solution

Understand the clock function

The given function \&\(x\) represents a clock function that cycles values between 0 and 12. For example, &\(13\) returns 1 because after reaching 12, the function starts again from 0.

Evaluate &\(36\)

To find \&\(36\), determine how many complete cycles of 12 fit into 36. Since 36 ÷ 12 = 3, 36 fits exactly into three complete cycles, resulting in the remainder of 0. Thus, \&\(36\)= 0.

Evaluate &\(12\)

Since 12 fits exactly into one complete cycle, \&\(12\)= 0. The clock function resets every 12 steps, so the function returns 0 at 12.

Divide &\(36\) by &\(12\)

Now, divide the results obtained: \[\frac{\&(36)}{\&(12)} = \frac{0}{0} \]. Therefore, this division is undefined as division by zero is undefined.

Reassess the possible mistake

Since both values of &\(36\) and &\(12\) yield 0 and division by zero is not valid, reassess the interpretation or consider if there might be additional context. Given the choices, note a misunderstanding in the function's cycling values.

Key concepts

Modular Arithmetic in GMAT

Modular arithmetic, often referred to as 'clock arithmetic,' is a crucial concept in mathematics and is commonly featured in GMAT problems. This system operates by wrapping around after reaching a certain value, akin to a clock resetting after 12 hours. When we say that the function \(\text{\textbackslash&}(x)\) returns values between 0 and 12, we are employing modular arithmetic with a modulus of 12. For example, \(\text{\textbackslash&}(13)\) equals 1 because, after reaching 12, we start again at 0. In the given problem, we apply this understanding to find \(\text{\textbackslash&}(36)\) and \(\text{\textbackslash&}(12)\). By consistently dividing 36 by 12, the remainder is zero, leading to \(\text{\textbackslash&}(36) = 0\). Similarly, \(\text{\textbackslash&}(12) = 0\) because 12 divided by 12 also leaves a remainder of zero. Understanding modular arithmetic helps simplify complex GMAT problems and ensures you handle cycles effectively.

Function Evaluation in Standardized Tests

Function evaluation is a fundamental skill tested in many standardized exams, including the GMAT. These problems usually involve understanding given functions and applying them to specific inputs to determine the output. In the clock function problem, the function \(\text{\textbackslash&}(x)\) is defined such that it cycles values between 0 and 12. This cyclic behavior is evaluated to find the results for specific inputs like \(\text{\textbackslash&}(36)\) and \(\text{\textbackslash&}(12)\). By breaking down the function step-by-step, you can effectively solve these questions. For instance, evaluating \(\text{\textbackslash&}(36)\) involves determining how many times 12 fits into 36 and seeking the remainder - which is zero in this case. This method also applies when finding \(\text{\textbackslash&}(12)\). Hence, function evaluation requires understanding the rules governing the function and applying them sequentially.

Division by Zero in Mathematics

Division by zero is a fundamental concept that often confuses students. In mathematics, dividing a number by zero is undefined. This undefined behavior arises because there is no number that, when multiplied by zero, can yield a finite number. In the presented problem, evaluating \(\text{\textbackslash&}(36)\) and \(\text{\textbackslash&}(12)\) both yield 0. Attempting to divide \(0 \div 0\) results in an undefined expression. It's crucial to recognize when division by zero appears in problems. Understanding that such operations are undefined helps prevent mathematical errors and ensures accurate solutions. Always remember to reassess the problem if you encounter division by zero, as it indicates a need for reevaluation.