Factors, Multiples & Prime Factorization Cheat Sheet

Factors, multiples, and prime factorization help students understand how numbers are built and related. This cheat sheet is useful for simplifying fractions, comparing numbers, finding common denominators, and solving word problems. It gives grades 4 through 8 students a clear reference for the number facts and strategies they use again and again.

A factor divides a number evenly, while a multiple is the result of multiplying a number by a whole number. A prime number has exactly $2$ factors, $1$ and itself, and a composite number has more than $2$ factors. Prime factorization breaks a composite number into prime factors, such as $36 = 2^2 \cdot 3^2$ , which can be used to find the greatest common factor and least common multiple.

Key Facts

A factor of $n$ is a whole number that divides $n$ with no remainder, such as $6$ being a factor of $24$ because $24 \div 6 = 4$ .
A multiple of $n$ is any number in the pattern $n \times 0, n \times 1, n \times 2, n \times 3, \ldots$ .
A prime number has exactly $2$ factors, $1$ and itself, such as $2, 3, 5, 7, 11$ .
A composite number has more than $2$ factors, such as $12$ because its factors are $1, 2, 3, 4, 6, 12$ .
The prime factorization of a number writes it as a product of primes, such as $60 = 2^2 \cdot 3 \cdot 5$ .
The greatest common factor, or GCF, is the largest factor shared by two or more numbers, such as $\operatorname{GCF}(18, 24) = 6$ .
The least common multiple, or LCM, is the smallest positive multiple shared by two or more numbers, such as $\operatorname{LCM}(6, 8) = 24$ .
Using prime factorization, the GCF uses the lowest shared powers of primes, while the LCM uses the highest powers of all primes.

Vocabulary

Factor: A factor is a whole number that divides another whole number evenly with no remainder.
Multiple: A multiple is the product of a number and a whole number, such as $4, 8, 12, 16$ for multiples of $4$ .
Prime Number: A prime number is a whole number greater than $1$ with exactly $2$ factors, $1$ and itself.
Composite Number: A composite number is a whole number greater than $1$ that has more than $2$ factors.
Prime Factorization: Prime factorization is writing a number as a product of prime numbers.
Greatest Common Factor: The greatest common factor is the largest whole number that divides each number in a set evenly.

Common Mistakes to Avoid

Confusing factors with multiples is a common mistake because factors are usually less than or equal to the number, while multiples are usually greater than or equal to the number.
Calling $1$ a prime number is wrong because a prime number must have exactly $2$ factors, and $1$ has only $1$ factor.
Stopping a prime factorization too soon is wrong because every factor in the final answer must be prime, such as writing $48 = 6 \cdot 8$ instead of $48 = 2^4 \cdot 3$ .
Forgetting repeated prime factors can change the answer, such as writing $72 = 2 \cdot 3^2$ instead of $72 = 2^3 \cdot 3^2$ .
Using the GCF when the problem asks for the LCM is wrong because the GCF finds the largest shared factor, while the LCM finds the smallest shared multiple.

Practice Questions

1 List all factors of $36$ .
2 Find the prime factorization of $84$ using exponents.
3 Find $\operatorname{GCF}(30, 45)$ and $\operatorname{LCM}(30, 45)$ .
4 A student says every multiple of $6$ is also a factor of $6$ . Explain why this statement is incorrect.

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Factors, Multiples & Prime Factorization Cheat Sheet

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