Let \phi(n) be the Euler Phi Function. If 1 \leq n \leq 1000, what is the smallest integer value of n that minimizes \frac{\phi(n)}{n}?
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To find the smallest integer value of n that minimizes the ratio φ(n) / n, we need to calculate the Euler Phi Function φ(n) for each integer 1 ≤ n ≤ 1000 and find the value of n that gives the smallest ratio φ(n) / n.
The Euler Phi Function φ(n) counts the number of positive integers less than or equal to n that are co-prime with n (i.e., the greatest common divisor of those integers with n is 1).
Here's how you can calculate the Euler Phi Function for each integer 1 ≤ n ≤ 1000:
1. Initialize a variable, let's say min_ratio, to a very large value (e.g., infinity) and a variable min_n to zero.
2. Iterate over each integer n from 1 to 1000.
3. For each n, calculate the Euler Phi Function φ(n) by counting the number of positive integers less than or equal to n that are co-prime with n.
- Start by initializing a variable, let's say count, to 0.
- Iterate over each positive integer k from 1 to n.
- If the greatest common divisor of k and n is 1 (i.e., gcd(k, n) = 1), increment the count by 1.
- After iterating over all the integers from 1 to n, φ(n) will be equal to the count.
4. Calculate the ratio φ(n) / n for each n.
- Compute the value of φ(n) / n.
5. If the ratio φ(n) / n is smaller than min_ratio, update min_ratio to the new ratio and min_n to the current value of n.
6. Once you have checked all the integers from 1 to 1000, the value of min_n will give you the smallest integer that minimizes φ(n) / n.
Let's follow these steps to find the answer.