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Using divisibility properties, can you find the total number of factors of \( 422851 \)?

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To find the total number of factors of \( 422851 \), we start by factoring it into its prime components and then using the properties of divisors.

1. **Prime Factorization of \( 422851 \)**:
   Let’s break down \( 422851 \) by checking for divisibility by small prime numbers.

   After testing, we find:
   \[
   422851 = 29 \times 14581
   \]
   Then, we further factor \( 14581 \):
   \[
   14581 = 29 \times 503
   \]
   Thus:
   \[
   422851 = 29^2 \times 503
   \]
   So, the prime factorization of \( 422851 \) is \( 29^2 \times 503^1 \).

2. **Calculating the Total Number of Divisors**:
   For a number of the form \( p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_n^{e_n} \), the total number of divisors is given by:
   \[
   (e_1 + 1)(e_2 + 1) \cdots (e_n + 1)
   \]
   Applying this to our prime factorization \( 29^2 \times 503^1 \):
   \[
   (2 + 1)(1 + 1) = 3 \times 2 = 6
   \]

3. **Conclusion**:
   The total number of factors of \( 422851 \) is \( 6 \).

Would you like a more detailed breakdown of each step?

---

1. What are the divisors of \( 422851 \) specifically?
2. How can we generalize this process for any composite number?
3. What is the significance of unique prime factorization in finding factors?
4. How do divisibility rules help in the initial factorization?
5. Can we determine the sum of all factors for \( 422851 \) as well?

**Tip:** Remember, each exponent in the prime factorization is increased by 1 to calculate divisors.

Using divisibility properties, can you find the total number of factors of 422851?

Discover how to calculate the total number of factors for 422851 by using prime factorization and divisibility properties. This detailed guide covers core concepts in number theory, including the prime factorization theorem and divisor-counting formula, ideal for high school students.

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