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.

Solution:

Array means arrangement of numbers in terms of rows and columns.

The conjecture is not True.

So, the conjecture made by chloe that given any two numbers, the greater number can always be arranged into more arrays is not true.

For example,

1. (2,3)

2→ (2*1), (1*2)

3→ (3*1), (1*3)

This pair has same number of arrays because both of them are prime.

2. (3,4)

Here, 4>3

3→ (3 * 1), (1*3)

4→ (4*1), (1*4), (2*2)

The pair of numbers are Co-prime, one of them is not prime. So Composite number has more number of arrays.

4. (6,11)

6→ (1*6), (6*1), (2*3), (3*2)

11→ (1*11), (11*1)

The number 6 is smaller, but it has more number of arrays.

5. (6,8)

6→ (1*6), (6*1), (2*3), (3*2)

9→ (1*9), (9*1), (3*3)

Both the numbers are composite. Smaller number 6 has more arrays.

These examples disprove the Conjecture made by Chloe, which states that given any two numbers, the greater number can always be arranged into more arrays.