Fractions Equal to Integers

A fraction represents an integer precisely when it can be written in the form $$ \frac{a}{b} = c \in Z $$ with a,b,c∈Z and b≠0. This occurs exactly when $$ a = b \cdot c $$ so that $$ \frac{a}{b} = \frac{b \cdot c}{b} = c $$ Here Z denotes the set of integers.

Such fractions belong to the class of equivalent fractions and are characterized by being equivalent to a fraction whose denominator is equal to one.

More generally, every integer can be expressed as a fraction, and a fraction represents an integer if and only if its numerator is divisible by its denominator.

Interpretation. Although written in fractional form, these expressions represent whole numbers. This happens precisely when the division of the numerator by the denominator is exact, that is, when no remainder occurs.

How to recognize such fractions

A fraction $$ \frac{a}{b} $$ with a,b∈Z and b≠0 represents an integer if and only if b divides a.

$$ b \mid a $$

Equivalently, there exists an integer k∈Z such that

$$ a = b \cdot k $$

and therefore

$$ \frac{a}{b} = k $$

Example. Consider the fraction $$ \frac{6}{3} $$ Since 3 divides 6, we can write 6 = 3·2. Hence $$ \require{cancel} \frac{6}{3} = \frac{ 3 \cdot 2 }{ 3 } = \frac{ \cancel{3} \cdot 2}{ \cancel{3} } = 2 $$ After factoring both the numerator and the denominator and simplifying the common factors, we obtain an integer.

Example

Consider the fraction

$$ \frac{6}{3} $$

The greatest common divisor (GCD) of the numerator and the denominator is 3.

$$ MCD(6,3) = 3 $$

Dividing both the numerator and the denominator by their greatest common divisor, we obtain an equivalent fraction in lowest terms

$$ \frac{6}{3} = \frac{ \frac{6}{3} }{ \frac{3}{3} } $$

$$ \frac{6}{3} = \frac{2}{1} $$

Since any fraction with denominator equal to one represents an integer, it follows that

$$ \frac{6}{3} = 2 $$

Therefore, the fraction 6/3 represents the integer 2.

Further examples

The following fractions represent integers

$$ \frac{12}{4} = 3 $$

$$ \frac{33}{3} = 11 $$

$$ \frac{4}{4} = 1 $$

$$ \frac{8}{4} = 2 $$

$$ \frac{-6}{6} = -1 $$

Characterization

A fraction represents an integer if and only if its denominator divides its numerator $$ \forall \ a,b \in Z \ , \ b \ne 0 \Rightarrow \left( \frac{a}{b} \in Z \Longleftrightarrow b \mid a \right) $$

Let a,b∈Z with b≠0 and consider the fraction

$$ \frac{a}{b} $$

If b divides a, then there exists k∈Z such that a = b·k, and therefore

$$ \frac{a}{b} = k \in Z $$

Conversely, if $$ \frac{a}{b} \in Z $$ then there exists k∈Z such that $$ \frac{a}{b} = k $$ hence a = b·k and therefore b divides a.

Thus, a fraction represents an integer if and only if its denominator divides its numerator.