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Use the image to help you answer Parts A and B.

Part A

Eggs are generally sold in batches of 12 (or dozens). The image shows the 3 different ways that 12 eggs can be packaged into arrays. Note that the array 3 by 4 is taken to be equivalent to the array 4 by 3 and therefore is not included.

If it was decided that eggs should be sold in batches of 24, then these 24 eggs could be packaged into a total of
different arrays.

Question 2
Three arrays are shown, each with 12 equal-sized cells. The first array has 2 rows and 6 columns. The second array has 1 row and 12 columns. The third array has 4 rows and 3 columns.

Part B
Chloe made a conjecture that given any two numbers, the greater number can always be arranged into more arrays.

State whether you agree or disagree. Then explain why you think so by giving an example of two numbers that prove or disprove the conjecture.

(2 points)

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1 answer

Part A

To find the different ways to package 24 eggs into arrays, we start by determining the factor pairs of 24. The factor pairs correspond to the number of rows and columns which form the arrays.

The factors of 24 are:

  • 1 x 24
  • 2 x 12
  • 3 x 8
  • 4 x 6

We also consider the arrays that are equivalent when swapped (e.g., 3 x 8 is the same as 8 x 3), so we only list the unique arrangements:

  • 1 row and 24 columns (1 x 24)
  • 2 rows and 12 columns (2 x 12)
  • 3 rows and 8 columns (3 x 8)
  • 4 rows and 6 columns (4 x 6)

Thus, the unique arrays of packaging 24 eggs into arrays give us:

  1. 1 x 24
  2. 2 x 12
  3. 3 x 8
  4. 4 x 6
  5. 6 x 4
  6. 8 x 3
  7. 12 x 2
  8. 24 x 1

However, since the task is to count unique layouts without equivalent arrangements, we count only the first four pairs:

  • 1 x 24
  • 2 x 12
  • 3 x 8
  • 4 x 6

Therefore, there are 4 distinct ways to package 24 eggs into arrays.

Part B

I disagree with Chloe's conjecture that "given any two numbers, the greater number can always be arranged into more arrays."

To illustrate this point, let's consider the numbers 6 and 8:

  • The factor pairs for 6 (which has factors of 1, 2, 3, and 6) are:

    • 1 x 6
    • 2 x 3

    So, 6 can be arranged into 2 unique arrays.

  • The factor pairs for 8 (which has factors of 1, 2, 4, and 8) are:

    • 1 x 8
    • 2 x 4

    So, 8 can be arranged into 2 unique arrays as well.

In this example, both 6 and 8 can be arranged into 2 arrays despite the fact that 8 is greater than 6. Thus, this example serves to disprove Chloe's conjecture that the greater number can always be arranged into more arrays.