Let's analyze the problem by understanding the characteristics of numbers in the form of [tex]\(2^k\)[/tex], where [tex]\(k\)[/tex] is an arbitrary integer.
Step 1: Express the Number
The number given is [tex]\(2^k\)[/tex]. For example:
- [tex]\(2^1 = 2\)[/tex]
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(2^3 = 8\)[/tex]
- And so on.
Step 2: Identify Properties of [tex]\(2^k\)[/tex]
Numbers in the form [tex]\(2^k\)[/tex] are powers of 2. [tex]\(2^k\)[/tex] is constituted solely of the factor 2 and involves repeated multiplication of the number 2.
Step 3: Consider What an Odd Divisor Is
An odd divisor is a divisor that is not divisible by 2. In other words, it has no factor of 2.
Step 4: Determine Odd Divisors of [tex]\(2^k\)[/tex]
Since [tex]\(2^k\)[/tex] is made up entirely of the factor 2:
- It is an even number for all [tex]\(k \geq 1\)[/tex].
- The only possible odd number is 1.
Step 5: Conclusion
The only odd divisor of [tex]\(2^k\)[/tex] (since it only has the factor 2) is the number 1. This is because 1 is the only number that is a divisor of every integer and is odd.
Thus, the number of odd divisors of [tex]\(2^k\)[/tex] is:
Answer: 1
Therefore, the correct answer is:
b) 1
