A student asks if $\frac{0}{6}$ is in its simplest form. What is the correct response?

A$\frac{0}{6}$ is in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) < 1$. Notice that $\operatorname{GCD}(0, 6) = 0$.
B$\frac{0}{6}$ is in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) = 1$. Notice that $\operatorname{GCD}(0, 6) = 1$.
C$\frac{0}{6}$ is not in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) = 1$. Notice that $\operatorname{GCD}(0, 6) = 0$. The simplest form of $\frac{0}{6}$ is $\frac{0}{0}$.
D$\frac{0}{6}$ is not in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) = 1$. Notice that $\operatorname{GCD}(0, 6) = 6$. The simplest form of $\frac{0}{6}$ is $\frac{0}{1}$.

Question

01-11-2026
Mathematics

Answered

A student asks if $\frac{0}{6}$ is in its simplest form. What is the correct response?

A$\frac{0}{6}$ is in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) < 1$. Notice that $\operatorname{GCD}(0, 6) = 0$.
B$\frac{0}{6}$ is in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) = 1$. Notice that $\operatorname{GCD}(0, 6) = 1$.
C$\frac{0}{6}$ is not in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) = 1$. Notice that $\operatorname{GCD}(0, 6) = 0$. The simplest form of $\frac{0}{6}$ is $\frac{0}{0}$.
D$\frac{0}{6}$ is not in simplest form. A fraction $\frac{a}{b}$ is in simplest form if, and only if, $b > 0$ and $\operatorname{GCD}(a, b) = 1$. Notice that $\operatorname{GCD}(0, 6) = 6$. The simplest form of $\frac{0}{6}$ is $\frac{0}{1}$.

Verified Answer :

Related Questions

Does the equator cross the northern or southern part of south america South america is located mostly on which two hemispheres

What is the main fuel consumed in the core of a red giant a H b C c Fe d He

your checking account is overdrawn by 45you write a check for 30then you make deposit of 100what I the balance in your account

How were the Mycenaeans and the Minoans similar

what times what give you 16

Which figure emerged from the August Coup as a national hero in Russia

3 Discuss Thomas Jeffersons views on democracy and liberty ashighlighted in the transcript

Cotton used to make clothes shoes blankets etc comes from a plant True False

How is a compound sentence formed A By joining a dependent and independent clause with a conjunction B By joining a simple sentence with a dependent clause C By

What do dragonflies eat

GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T21:30:20-05:00

Define the simplest form of a fraction $\frac{a}{b}$ as requiring $b > 0$ and $\operatorname{GCD}(|a|, b) = 1$.
Calculate $\operatorname{GCD}(0, 6)$: since all divisors of 6 are divisors of 0, the largest divisor of 6, which is 6, is the GCD.
Observe that $\operatorname{GCD}(0, 6) = 6 eq 1$, confirming that $\frac{0}{6}$ is not in its simplest form.
Divide both terms by the GCD to find the simplest form: $\frac{0 \div 6}{6 \div 6} = \frac{0}{1}$.
Conclude that $\boxed{\text{D}}$ is the correct response.

Explanation

Definition of a Fraction in Simplest Form

To determine if the fraction $\frac{0}{6}$ is in its simplest form, we first recall the definition of a fraction in simplest form. A fraction $\frac{a}{b}$ is considered to be in simplest form if the denominator $b$ is greater than zero and the greatest common divisor (GCD) of the numerator $a$ and the denominator $b$ is equal to $1$, i.e., $\operatorname{GCD}(|a|, b) = 1$.

Calculating the GCD of 0 and 6

Next, we examine the given fraction $\frac{0}{6}$. Here, $a = 0$ and $b = 6$. To check if it is in simplest form, we calculate the greatest common divisor of $0$ and $6$. The divisors of $6$ are $1, 2, 3, \text{and } 6$. Since any non-zero integer $n$ divides $0$ (as $0 = n \times 0$), all divisors of $6$ are also divisors of $0$. Therefore, the greatest common divisor $\operatorname{GCD}(0, 6)$ is the largest divisor of $6$, which is $6$.

Simplifying the Fraction

Since $\operatorname{GCD}(0, 6) = 6$, which is not equal to $1$, the fraction $\frac{0}{6}$ does not satisfy the condition for being in its simplest form. To simplify it, we divide both the numerator and the denominator by their greatest common divisor, $6$: $\frac{0 \div 6}{6 \div 6} = \frac{0}{1}$. Thus, the simplest form of $\frac{0}{6}$ is $\frac{0}{1}$, which evaluates to $0$.

Conclusion and Final Answer

Comparing our findings with the given options, we see that Option D correctly identifies that $\frac{0}{6}$ is not in simplest form, states the correct condition for simplest form ($b > 0$ and $\operatorname{GCD}(a, b) = 1$), correctly calculates $\operatorname{GCD}(0, 6) = 6$, and provides the correct simplest form $\frac{0}{1}$. Therefore, the correct response is D.

Examples

The concept of the greatest common divisor (GCD) is essential for simplifying fractions to ensure consistency in mathematical communication. Just as we simplified $\frac{0}{6}$ to $\frac{0}{1}$ to recognize it as 0, this principle applies to any ratio or measurement. For instance, if a manufacturing process yields 0 defective items out of 6 total products, expressing this as a ratio $\frac{0}{1}$ standardizes the result, making it immediately clear that the defect rate is zero. Applying $\gcd(a, b) = 1$ helps in various fields, from scaling recipes to optimizing computer algorithms.