The number 2 is chosen to begin a ladder diagram to find the prime factorization of 66. What other numbers could have been used to start the ladder diagram for 66? How does starting with a different number change the diagram?
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The numbers are 1, 3, 6, 11, 22, and 33. It changes the diagram because it puts the prime factors in a different order, however the multiplication of the numbers in a different order does NOT change the product in any way. Starting the diagram with 2 still comes out to 66, as does starting the diagram with 3. 2•3•11=66. 3•2•11=66
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To find the prime factorization of 66 using a ladder diagram, we start by dividing the number by its smallest prime factor. In this case, since 2 is chosen as the starting point, we divide 66 by 2.
If we were to choose a different number to start the ladder diagram for 66, it must be a prime number that is a factor of 66. The prime factorization of 66 is calculated as follows:
1. 66 ÷ 2 = 33
2. 33 ÷ 3 = 11
Starting with a different number changes the diagram as it alters the order in which we divide the number. For example, if we started with a different prime number, such as 3 instead of 2, the diagram would look like:
1. 66 ÷ 3 = 22
2. 22 ÷ 2 = 11
As you can see, even though we started with a different number, the end result is the same – the prime factorization is 2 * 3 * 11.
So, the other numbers that could have been used to start the ladder diagram for 66 would be any of the prime factors of 66, namely 2, 3, and 11.