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I am an AI language model developed by OpenAI. I don't have a personal name, but you can refer to me as "Assistant". How can I help you today?
Eggs are generally sold in batches of 12 (or dozens). The image shows the three 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
To determine the number of different arrays in which 24 eggs can be packaged, we can use the concept of divisors or factors of a number.
The divisors for 24 are: 1, 2, 3, 4, 6, 8, 12, and 24. These are all the possible numbers by which 24 can be divided evenly.
Since we want to find the different arrays, we need to consider the dimensions of the arrays. For example, if we have an array of 4 by 6 (row by column), it is equivalent to an array of 6 by 4. So, to avoid duplication, we only need to count one of these arrangements.
The divisors of 24 can be paired up with their corresponding divisors as follows:
1 x 24
2 x 12
3 x 8
4 x 6
Therefore, there are a total of 4 different arrays in which 24 eggs can be packaged.
Chole 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.
I disagree with Chole's conjecture. There can be cases where the greater number does not necessarily result in more arrays. Let's consider an example:
Take the numbers 10 and 16. The divisors of 10 are 1, 2, and 5, while the divisors of 16 are 1, 2, 4, 8, and 16.
For the number 10, we can have the following arrays:
1 by 10
2 by 5
For the number 16, we can have the following arrays:
1 by 16
2 by 8
4 by 4
In this case, we can see that even though 16 is the greater number, the number of arrays it can be arranged into is fewer than the number of arrays that 10 can be arranged into. Therefore, this example disproves Chole's conjecture that the greater number can always be arranged into more arrays.
Find the first 3 composite numbers that are multiples of 4 and greater than 10.
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
The three numbers are_______________
To find the first 3 composite numbers that are multiples of 4 and greater than 10, let's examine the given numbers:
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Out of these numbers, we need to identify which ones are composite (numbers greater than 1 that have factors other than 1 and themselves) and multiples of 4.
The composite numbers in the given list are:
12, 14, 15, 16, 18, and 20.
Out of these, the numbers that are also multiples of 4 are:
12, 16, and 20.
Therefore, the first three composite numbers that are multiples of 4 and greater than 10 are: 12, 16, and 20.