To find the number of positive integers n for which 1/n yields a terminating decimal with a non-zero hundredths digit, we need to understand the conditions under which a fraction will terminate.
A fraction in its simplest form will terminate if its denominator has prime factors only of 2 and/or 5. This is because other prime factors in the denominator would result in a recurring decimal.
Let's analyze the prime factors of n:
- If n has only prime factors of 2 and/or 5, then the decimal representation of 1/n will terminate.
- If n has any prime factors other than 2 and 5, then the decimal representation of 1/n will be recurring.
Now, let's consider the condition of having a non-zero hundredths digit. A terminating decimal with a non-zero hundredths digit means that the denominator n should have at least one prime factor other than 2 and 5.
From this analysis, we can conclude that the number of positive integers n where 1/n yields a terminating decimal with a non-zero hundredths digit is equal to the number of positive integers that have prime factors other than 2 and 5.
To determine this count, we can use the concept of complementary counting. We can find the total count of positive integers that only have prime factors of 2 and 5 and subtract it from the total count of positive integers.
Since there are infinitely many positive integers, the total count of positive integers is infinite. However, for practical purposes, we can define a range or limit based on the context of the problem.
For example, if we want to find the count of positive integers n up to a certain value such as 100, we can iterate through the numbers from 1 to 100 and check if each number has prime factors other than 2 and 5.
Alternatively, if we want to find the count of positive integers up to a certain power of 10, such as up to 10^6, we can use number theory techniques to calculate it more efficiently, such as counting the numbers in a range that have prime factors other than 2 and 5.
It is important to note that even though the total count of positive integers is infinite, the count of positive integers n with 1/n yielding a terminating decimal with a non-zero hundredths digit is finite within the given range or limit.
Overall, to determine the exact number of positive integers n, you would need to define the range or limit for the analysis and employ number theory techniques to calculate it accurately.