Question

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.

Answer 1

I disagree with Chloe's conjecture that the greater number can always be arranged into more arrays than the smaller number.

To illustrate this, consider two numbers: 3 and 4. The number 4 can be arranged into an array in various ways, such as:

- 1 row of 4 (4)
- 2 rows of 2 (2, 2)
- 4 rows of 1 (1, 1, 1, 1)

Now, the number 3 can also be arranged into arrays:

- 1 row of 3 (3)
- 3 rows of 1 (1, 1, 1)
- 1 row of 3 can also be represented as 3 columns of 1 (1, 1, 1)

However, in terms of the number of distinct rectangular arrangements, we can identify the number of ways to arrange each number:

- The number 3 has 2 distinct arrangements: (1x3) and (3x1).
- The number 4 has 3 distinct arrangements: (1x4), (2x2), and (4x1).

In this case, 4 has more arrangements than 3. However, if we consider the number 1, which can only be arranged as 1 (1x1), it has fewer arrangements than 2 (which can be arranged as 2x1 and 1x2).

Thus, the conjecture does not hold true for every pair of numbers, specifically as we can find examples where the smaller number has fewer distinct arrangements than the larger one could inherently allow, or other configurations based on the spread of factors lead to fewer arrangements overall.