To find the number of divisors of [tex]\(2^{2009}\)[/tex], we can follow these steps:
1. Understanding the Structure of the Number:
- The number [tex]\(2^{2009}\)[/tex] is a power of 2. Its prime factorization consists of a single prime factor, 2, raised to the power of 2009.
2. Formula for the Number of Divisors:
- For any given number [tex]\( n \)[/tex] with a prime factorization of the form [tex]\( p_1^{e1} \cdot p_2^{e2} \cdot ... \cdot p_k^{ek} \)[/tex], the number of divisors [tex]\( D(n) \)[/tex] can be determined using the formula:
[tex]\[
D(n) = (e1 + 1)(e2 + 1)...(ek + 1)
\][/tex]
where [tex]\( e1, e2, ..., ek \)[/tex] are the exponents of the prime factors.
3. Apply the Formula to [tex]\(2^{2009}\)[/tex]:
- In our case, the prime factorization of [tex]\(2^{2009}\)[/tex] is simply [tex]\(2^{2009}\)[/tex].
- Here, we have only one prime factor (2) with exponent 2009.
4. Calculate the Number of Divisors:
- According to the formula, the number of divisors [tex]\( D(2^{2009}) \)[/tex] will be:
[tex]\[
D(2^{2009}) = 2009 + 1
\][/tex]
- Therefore,
[tex]\[
D(2^{2009}) = 2010
\][/tex]
Thus, [tex]\(2^{2009}\)[/tex] has 2010 divisors, including 1 and the number itself.
