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Roman knows that 6 and 7 are a factor pair of 42, because $42=6 \times 7$. Roman found another factor pair of 42 using the steps shown.



Choose a description for each step Roman used.

$\begin{aligned}

42 & =6 \times 7 \\

& =(2 \times 3) \times 7 \\

& =2 \times(3 \times 7) \\

& =2 \times 21

\end{aligned}$

Verified Answer :

  • The first step, \(6 \times 7 \rightarrow (2 \times 3) \times 7\), applies prime factorization to replace 6 with its factors 2 and 3.
  • The second step, \((2 \times 3) \times 7 \rightarrow 2 \times (3 \times 7)\), uses the associative property of multiplication to regroup the factors.
  • The final step, \(2 \times (3 \times 7) \rightarrow 2 \times 21\), uses simplification to evaluate the product inside the parentheses.
  • The resulting pair is \(\boxed{(2, 21)}\).

Explanation

  1. Initial State Analysis

The initial equation is \(42 = 6 \times 7\). This expresses 42 as a product of its factors. We are given a sequence of steps to transform this pair into another factor pair, (2, 21), and need to identify the mathematical principles applied at each step.

  1. Step 1 Analysis: Prime Factorization/Substitution

In the first step of the transformation, \(6 \times 7\) becomes \((2 \times 3) \times 7\). Here, the number 6 is replaced by its prime factors 2 and 3. This is an application of prime factorization or substitution of a factor with its components. Since \(6 = 2 \times 3\), substituting this value preserves the product.

  1. Step 2 Analysis: Associative Property of Multiplication

In the second step of the transformation, \((2 \times 3) \times 7\) becomes \(2 \times (3 \times 7)\). This shift of parentheses is a direct application of the associative property of multiplication, which states that for any numbers \(a\), \(b\), and \(c\), \((a \times b) \times c = a \times (b \times c)\). This property allows us to regroup factors without changing the result of the product.

  1. Step 3 Analysis: Simplification

In the final step of the transformation, \(2 \times (3 \times 7)\) becomes \(2 \times 21\). This is done by simplifying the expression inside the parentheses: calculating \(3 \times 7 = 21\). Performing the multiplication within the grouped terms is simply arithmetic simplification.

  1. Final Conclusion

By analyzing the steps, we have identified the progression: first, prime factorization of 6 to (2 x 3); second, regrouping using the associative property to (2 x (3 x 7)); and third, simplification to 2 x 21. This confirms that (2, 21) is indeed a valid factor pair for 42. Final summary: The steps are Substitution/Prime Factorization, Associative Property, and Simplification.

Examples

Understanding factor pairs and regrouping is a fundamental skill in number theory and algebra. For instance, if you are planning to arrange 42 chairs in a room, you might start with 6 rows of 7 chairs. If you decide that 6 rows are too many and you want only 2 rows, you can use these exact steps—factorizing 6 into 2 x 3, regrouping 3 x 7, and finding that you need 21 chairs per row—to determine your new layout (2 rows of 21 chairs each). This technique is widely used in organizing data, tiling surfaces, or managing resources efficiently.

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