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art 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.

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.

1 answer

To address Part A, let's first determine how many different arrays can be created using 24 eggs. The goal is to find all the unique ways to arrange 24 as a product of two integers, where the order does not matter.

The factors of 24 are:

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

Now, we need to consider only the unique pairs (since the order doesn't matter). The unique arrangements are:

  1. (1, 24)
  2. (2, 12)
  3. (3, 8)
  4. (4, 6)

Thus, the possible arrays for 24 eggs are:

  1. 1 row of 24 eggs
  2. 2 rows of 12 eggs
  3. 3 rows of 8 eggs
  4. 4 rows of 6 eggs
  5. 6 rows of 4 eggs (same as 4 rows of 6 eggs)
  6. 8 rows of 3 eggs (same as 3 rows of 8 eggs)
  7. 12 rows of 2 eggs (same as 2 rows of 12 eggs)
  8. 24 rows of 1 egg (same as 1 row of 24 eggs)

After eliminating duplicates due to order:

  • (1, 24)
  • (2, 12)
  • (3, 8)
  • (4, 6)

So, the number of unique arrays for 24 eggs is 4.

Now, for Part B, I disagree with Chloe's conjecture that a greater number can always be arranged into more arrays.

Here's a counter-example: Let's consider the numbers 6 and 8.

For 6:

  • The factors are:
    • 1 x 6
    • 2 x 3

So, the unique arrays are:

  1. (1, 6)
  2. (2, 3)

This gives us 2 unique arrays.

For 8:

  • The factors are:
    • 1 x 8
    • 2 x 4
    • 2 x 4 (already counted considering order does not matter)

So, the unique arrays are:

  1. (1, 8)
  2. (2, 4)

This gives us 2 unique arrays for 8 as well.

In this case, both 6 and 8 can be arranged into the same number of unique arrays (2). Therefore, Chloe's conjecture is disproven by this example because having a greater number does not guarantee more unique arrangements.