To determine how many natural numbers between 1 and 100 are divisible by 5, we need to identify all the multiples of 5 within this range. Let's break down the process step-by-step:
1. Define the Range: We are looking at natural numbers from 1 to 100.
2. Identify the Multiples of 5: A natural number [tex]\( n \)[/tex] is divisible by 5 if it can be expressed as [tex]\( n = 5k \)[/tex] where [tex]\( k \)[/tex] is an integer.
3. Find the Smallest Multiple of 5 within the Range: The smallest multiple of 5 within our range is 5 itself (since 5 1 = 5).
4. Find the Largest Multiple of 5 within the Range: The largest multiple of 5 within our range is 100 (since 5 20 = 100).
5. Count the Multiples of 5: We now need to count all integers [tex]\( k \)[/tex] such that 1 ≤ [tex]\( 5k \)[/tex] ≤ 100.
- The first value of [tex]\( k \)[/tex] is 1 (giving us 5).
- The second value of [tex]\( k \)[/tex] is 2 (giving us 10).
- This continues up to the value of [tex]\( k = 20 \)[/tex] (giving us 100).
6. Conclusion: There are 20 integers [tex]\( k \)[/tex] that produce values [tex]\( 5k \)[/tex] within the range 1 to 100.
Thus, the number of natural numbers between 1 and 100 that are divisible by 5 is:
[tex]\[ \boxed{20} \][/tex]
Given the options:
(a) 197
(b) 198
(c) 199
(d) 200
None of the given options are correct based on this explanation, the accurate answer derived is 20 natural numbers.
