Coprime Numbers: Guide and Examples

What Is Coprime Numbers?

To determine how to check coprime numbers, you must look at their individual factors.

Two numbers are coprime if no positive integer other than $1$ divides evenly into both of them.

These number pairs are also frequently called relatively prime integers or mutually prime numbers.

In the diagram, $8$ and $15$ are both composite numbers because they have several factors.

However, because their only shared factor is $1$, they are coprime to each other.

Key Ideas and Vocabulary

There are several reliable rules that can help you identify coprime numbers quickly.

• Consecutive integers: Any two consecutive integers, such as $14$ and $15$, are always coprime.
• Prime numbers: Any two different prime numbers are always coprime because their only factors are $1$ and themselves.
• The number 1: The integer $1$ is coprime to every other integer.

When analyzing properties like the GCF and LCM, coprime numbers have a unique relationship.

Because their greatest common factor is $1$, you cannot divide both numbers by any larger integer to simplify them.

Visual Explanation

Fractions provide a perfect visual representation of coprime numbers.

A fraction is in its simplest form if, and only if, its numerator and its denominator are coprime.

In the first fraction, $14$ and $21$ share a factor of $7$, meaning they are not coprime, so the fraction can be simplified.

In the second fraction, $14$ and $15$ share no factors other than $1$, meaning they are coprime, so the fraction is already in simplest form.

Worked Examples

These coprime numbers examples demonstrate how to verify relationships using different methods.

Example 1: Checking with factor lists

Question: Are the numbers $9$ and $28$ coprime?

Method:

1. List all factors of $9$.
2. List all factors of $28$.
3. Compare the lists to find the greatest common factor.

Factors of $9$: $1$, $3$, $9$

Factors of $28$: $1$, $2$, $4$, $7$, $14$, $28$

The only number that appears in both lists is $1$.

Answer: Yes, $9$ and $28$ are coprime.

Check: Neither number can be evenly divided by any prime number that divides the other.

Example 2: Identifying non-coprime numbers

Question: Are $15$ and $25$ coprime?

Method:

1. List the factors of $15$.
2. List the factors of $25$.
3. Compare the lists.

Factors of $15$: $1$, $3$, $5$, $15$

Factors of $25$: $1$, $5$, $25$

Both lists share the factor $5$, meaning their greatest common factor is $5$, not $1$.

Answer: No, $15$ and $25$ are not coprime.

Check: Since $15 \div 5 = 3$ and $25 \div 5 = 5$, both numbers are divisible by $5$.

Example 3: Using prime factorization

Question: Are $21$ and $40$ coprime?

Method:

1. Use the fundamental theorem of arithmetic to find the prime factorization of each number.
2. Check if they share any prime factors.

Prime factorization of $21$: $3 \times 7$

Prime factorization of $40$: $2 \times 2 \times 2 \times 5$

The prime factors for $21$ are $3$ and $7$. The prime factors for $40$ are $2$ and $5$.

They share no prime factors.

Answer: Yes, $21$ and $40$ are coprime.

Check: The greatest common factor of any two numbers with no shared prime factors is always $1$.

Common Mistakes and Non-Examples

A widespread mistake is assuming that a pair of coprime numbers must consist of actual prime numbers.

The word "prime" in "relatively prime" describes the relationship between the two numbers, not the numbers themselves.

For example, $8$ is composite and $9$ is composite, but together they are coprime because they share no common factors other than $1$.

Additionally, finding the least common multiple of two coprime numbers is straightforward: simply multiply them together. If they are not coprime, multiplying them will give a common multiple, but not the least one.

Non-Example: Numbers with shared prime factors

Consider $12$ and $18$. Both are divisible by $2$ and $3$.

Because their greatest common factor is $6$, they are a non-example of coprime numbers.

Real-World Connections

Coprime numbers are essential in mechanical engineering, especially when designing gears.

If a small gear with $12$ teeth drives a large gear, engineers often give the large gear a coprime number of teeth, such as $35$.

Because $12$ and $35$ are coprime, the exact same teeth will rarely meet, which prevents the gears from wearing out in one specific spot.

In computer science, coprime numbers form the foundation of encryption algorithms that secure digital communications. For extremely large numbers used in cryptography, computers often use the Euclidean algorithm to verify if two numbers are coprime quickly without listing every factor.