each of the natural numbers 2 through 100, inclusive, is factored in its prime factorization. how many factors of 5 are in the collection of factorizations?
To find the number of factors of 5 in the prime factorization of the numbers 2 through 100, we need to consider the multiples of 5 within that range.
We can use two approaches to solve this:
1. Multiples of 5: We need to count the numbers that are multiples of 5. Between 2 and 100, there are four sets of multiples of 5: {5, 10, 15, ..., 100}. We can generate this set by starting with 5 and repeatedly adding 5 until reaching 100. By doing this, we can count the numbers that contribute at least one factor of 5 in their prime factorization.
2. Prime factorization: We can calculate the prime factorization of each number from 2 to 100 and count the number of factors of 5. For example, the prime factorization of 15 is 3 * 5, which contains one factor of 5. By finding the prime factorization of each number, we can count the factors of 5.
Let's use the second approach, as it provides a more accurate and systematic method.
We need to calculate the prime factorization of each number from 2 to 100. For numbers that have a factor of 5 in their prime factorization, we increment a counter.
Steps:
1. Start with the counter set to zero.
2. Iterate through each number from 2 to 100.
3. For each number, check if it has a factor of 5 in its prime factorization using the modulo operator (%). If the number is divisible by 5 (i.e., number % 5 == 0), increment the counter by one.
4. After iterating through all the numbers, the value of the counter will give you the total number of factors of 5 in the collection of factorizations.
By following this approach, you will be able to calculate the exact number of factors of 5 in the prime factorization of the numbers from 2 to 100.