Lesson 2: Find Greatest Common Factor and Least Common Multiple

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$.
If a number $m$ is a multiple of $n$, then $m$ is divisible by $n$.
If $a \cdot b = m$, and both $a$ and $b$ are integers, then $a$ and $b$ are factors of $m$.
A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.
A composite number is a counting number greater than 1 that is not prime.
The prime factorization of a number is the product of prime numbers that equals the number.

Examples

• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, since these are the integers that multiply to give 30.
• 17 is a prime number because its only factors are 1 and 17. 25 is a composite number because it has a factor of 5 besides 1 and 25.
• The prime factorization of 42 is $2 \cdot 3 \cdot 7$. We can find this by breaking it down: $42 = 6 \cdot 7 = (2 \cdot 3) \cdot 7$.

Explanation

Think of factors as building blocks for numbers. Prime numbers are the most basic blocks. Prime factorization is breaking a number down into its unique set of prime building blocks. This is useful for simplifying fractions and other calculations.

Property

The greatest common factor, GCF, of two numbers is that common factor that is the largest. Once we have the prime factorization of two numbers, the GCF is the product of all the prime factors common to both numbers. When we have a prime as a multiple factor of both numbers, the GCF takes the smaller of the multiples.

Examples

• Find the GCF of 48 and 36. Factors of 48: {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}. Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}. The GCF is 12.
• Find the GCF of 60 and 84 using prime factors. $60 = 2 \cdot 2 \cdot 3 \cdot 5$. $84 = 2 \cdot 2 \cdot 3 \cdot 7$. The common factors are $2 \cdot 2 \cdot 3$, so the GCF is 12.
• Use the GCF to simplify the sum $28 + 42$. Factors of 28: {1, 2, 4, 7, 14, 28}. Factors of 42: {1, 2, 3, 6, 7, 14, 21, 42}. The GCF is 14. Or using prime factors, $28 = 2 \cdot 2 \cdot 7$, $42 = 2 \cdot 3 \cdot 7$, the common factors are $2 \cdot 7$, so the GCF is 14. Then $28 + 42 = 14 \times 2 + 14 \times 3 = 14(2+3) = 14 \times 5 = 70$.

Explanation

The GCF is the biggest number that divides evenly into two or more numbers. It's the king of all the common factors! We use it to simplify fractions and expressions to their simplest forms.