To determine how many integers between 1 and 100 (inclusive) are divisible by 5, we start by identifying the pattern of numbers that meet this criterion.
An integer is divisible by 5 if, when divided by 5, it results in an integer (i.e., no remainder).
Let’s consider the sequence:
[tex]\[ 5, 10, 15, 20, \ldots, 100 \][/tex]
This is an arithmetic sequence where:
- The first term ([tex]\(a\)[/tex]) is 5,
- The common difference ([tex]\(d\)[/tex]) is 5,
- The last term ([tex]\(l\)[/tex]) is 100.
To find the number of terms ([tex]\(n\)[/tex]) in this arithmetic sequence, we use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
[tex]\[ a_n = a + (n - 1)d \][/tex]
We set [tex]\(a_n\)[/tex] to the last term:
[tex]\[ 100 = 5 + (n - 1) \cdot 5 \][/tex]
Rearrange and solve for [tex]\(n\)[/tex]:
[tex]\[ 100 = 5 + 5(n - 1) \][/tex]
[tex]\[ 100 = 5 + 5n - 5 \][/tex]
[tex]\[ 100 = 5n \][/tex]
[tex]\[ n = 20 \][/tex]
Therefore, there are 20 integers between 1 and 100 (inclusive) that are divisible by 5.
So, the correct answer is:
OC) 20
