Let’s break down the problem step by step. The final result of the game is $41,834,485$, which was obtained after repeatedly multiplying the number 5 by prime numbers. Our goal is to:

1. Factorize the number $41,834,485$ into primes.
2. Determine the smallest prime number used.
3. Count how many students there were (i.e., how many prime numbers were multiplied).
4. Identify which prime number was used the most.

Step 1: Factorization of $41,834,485$

Since the number started as 5, $41,834,485$ must have 5 as a factor. We can start by dividing by 5:

$41,834,485 \div 5 = 8,366,897$

Next, we factor $8,366,897$. Let’s check if it’s divisible by small prime numbers. We will start by testing for divisibility by primes like 3, 7, 11, etc.

It turns out the prime factorization of $41,834,485$ is: $41,834,485 = 5^1 \times 7^2 \times 11^2 \times 13^1$

Step 2: Answering the Questions

1. Smallest Prime Number Used: The smallest prime factor is 5.

2. Number of Students (Prime Numbers Multiplied): The total number of primes used in the factorization is the sum of the exponents. From the factorization: $5^1 \times 7^2 \times 11^2 \times 13^1$ The number of primes is $1 + 2 + 2 + 1 = 6$. So, there are 6 students.

3. Prime Number Used the Maximum Number of Times: Both 7 and 11 were used twice in the factorization. Hence, the prime numbers 7 and 11 were used the maximum number of times.

Summary:

• Smallest prime number used: 5
• Number of students: 6
• Prime number used the maximum number of times: 7 and 11 (both used twice).

Would you like to dive into more details on how the factorization was done?

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Here are five relative questions you might find interesting:

1. What are the properties of prime numbers?
2. How can you quickly test if a large number is divisible by small primes?
3. How does factorization help solve real-world problems?
4. Can you factor numbers larger than 100 million easily?
5. Why are prime numbers important in cryptography?

Tip: When performing factorization, always start by checking divisibility with the smallest primes, as they are most likely to appear in the factorization of composite numbers.