Finding Pairs with a Given GCF
Finding pairs with a given GCF is a Grade 6 math skill in Big Ideas Math Advanced 1, Chapter 1: Numerical Expressions and Factors. Students work backwards from a given greatest common factor (GCF) to find pairs of numbers that share exactly that GCF, using the relationship between GCF and factor lists to construct appropriate number pairs.
Key Concepts
Property To find a pair of numbers with a specific greatest common factor (GCF), multiply the GCF by two numbers that are coprime. Two numbers are coprime if their only common factor is 1. If $g$ is the desired GCF, the pair can be found by calculating $(g \cdot x)$ and $(g \cdot y)$, where $\text{GCF}(x, y) = 1$.
Examples Which pair of numbers has a GCF of 7? The pair $(14, 21)$ has a GCF of 7. This is because $14 = 7 \cdot 2$ and $21 = 7 \cdot 3$, and the numbers 2 and 3 are coprime. Find a pair of numbers with a GCF of 12. Choose two coprime numbers, like 3 and 5. The pair is $(12 \cdot 3, 12 \cdot 5)$, which is $(36, 60)$. The GCF of 36 and 60 is 12.
Explanation This skill involves working backward from a known greatest common factor (GCF). To construct a pair of numbers with a specific GCF, you must ensure two conditions are met. First, both numbers in the pair must be multiples of the desired GCF. Second, the factors you multiply the GCF by must not share any common factors other than 1, otherwise, the GCF of the resulting pair would be larger than intended.
Common Questions
How do you find pairs of numbers with a given GCF?
Start with the given GCF. Multiply it by two different integers that share no common factor themselves. For example, for GCF = 4, multiply 4 by 1 and 4 by 3 to get 4 and 12 (GCF = 4). Or use 4 x 1 = 4 and 4 x 5 = 20.
What is the GCF?
The greatest common factor (GCF) is the largest number that divides evenly into two or more numbers. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18.
How do you verify that two numbers have a specific GCF?
Find the factors of both numbers, identify the common factors, and confirm the largest one matches the required GCF. Alternatively, use the Euclidean algorithm or prime factorization to find the GCF.
Where is this skill taught in Big Ideas Math Advanced 1?
Finding pairs with a given GCF is covered in Chapter 1: Numerical Expressions and Factors of Big Ideas Math Advanced 1, the Grade 6 math textbook.