I disagree with Chloe's conjecture that the greater number can always be arranged into more arrays than the smaller number. This assertion is not universally true, as the ability to arrange numbers into arrays depends on their factors and divisibility.
Example
Let's take two numbers: 6 and 4.
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The number 6: The factors of 6 are 1, 2, 3, and 6. We can arrange 6 into the following arrays:
- 1 row of 6 (1x6)
- 2 rows of 3 (2x3)
- 3 rows of 2 (3x2)
- 6 rows of 1 (6x1)
This gives us 4 ways to arrange the number 6.
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The number 4: The factors of 4 are 1, 2, and 4. We can arrange 4 into:
- 1 row of 4 (1x4)
- 2 rows of 2 (2x2)
- 4 rows of 1 (4x1)
This gives us 3 ways to arrange the number 4.
Conclusion
In this example, 6 (the greater number) can be arranged into 4 arrays, while 4 (the smaller number) can be arranged into only 3 arrays. However, if we change the numbers to 4 and 2:
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The number 4 can still be arranged in 3 ways (as previously shown).
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The number 2 can only be arranged as:
- 1 row of 2 (1x2)
- 2 rows of 1 (2x1)
This gives us only 2 ways to arrange the number 2.
So in this specific case, the greater number (4) can be arranged into more arrays than the smaller number (2).
However, if we consider different pairs (like comparing 8 and 6):
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The number 8 can be arranged in:
- 1 row of 8 (1x8)
- 2 rows of 4 (2x4)
- 4 rows of 2 (4x2)
- 8 rows of 1 (8x1)
That makes for 4 ways.
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The number 6, as previously calculated, yields 4 ways too.
In this case, both numbers have the same number of arrangements.
Thus, the conjecture does not hold in all cases, and the number of arrangements depends on the specific factors of the numbers involved, not solely on their relative size. Therefore, I believe the conjecture is false.
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