91Ó°ÊÓ

Positive integers are simply the set of all whole numbers greater than 0. If you think about the counting numbers you learned as a kid, those are positive integers—like 1, 2, 3, and so on. In our exercise, we are focusing on positive integers up to 100. This means we only consider numbers from 1 to 100, inclusive.
How does this help? Well, it gives us a clear boundary for our problem of finding which numbers are divisible by 5 or 7.

Divisibility refers to the ability of one number to be divided by another without leaving a remainder. For example, a number is divisible by 5 if, when you divide it by 5, the result is a whole number. In math terms, an integer n is divisible by 5 if there exists an integer k such that: \( n = 5k \)
In our problem, we need to find numbers from 1 to 100 that are divisible by both 5 and 7. We can list these numbers in sequences:
* Numbers divisible by 5 include 5, 10, 15, ..., 100
* Numbers divisible by 7 include 7, 14, 21, ..., 98
* Numbers divisible by both 5 and 7 (i.e., 35) include 35, 70.

The inclusion-exclusion principle helps us count the number of items in overlapping groups. It's crucial to avoid double-counting the items that are common to both groups. The formula for two sets A and B is:\[ |A \, \text{or} \, B| = |A| + |B| - |A \, \text{and} \, B| \]
In our exercise, we let A be the set of numbers divisible by 5 and B be the set of numbers divisible by 7. Using the inclusion-exclusion principle, we compute: \[ |A \, \text{or} \, B| = |A| + |B| - |A \, \text{and} \, B| \] \[ |A| = 20, \, |B| = 14, \, |A \, \text{and} \, B| = 2 \]
Hence, \[ |A \, \text{or} \, B| = 20 + 14 - 2 = 32 \]
That is, there are 32 numbers from 1 to 100 that are divisible by either 5 or 7.

Sequence counting involves listing all numbers in a range that fit a particular criterion. We count these numbers by figuring out the pattern they form. For instance, to count numbers divisible by 5 in the range from 1 to 100, we recognize that these numbers form the sequence: \( 5, 10, 15, ..., 100 \)
The general term of this arithmetic sequence can be given by: \[ a_n = 5n \]
To find how many terms are in this sequence, we solve for n: \[ 100 / 5 = 20 \]
Similarly, we do this for the sequence of numbers divisible by 7: \( 7, 14, 21, ..., 98 \)
The general term here is: \[ b_n = 7n \]
Solving for n gives: \[ 100 / 7 \approx 14 \]
This arithmetic sequence approach makes sequence counting an organized method for identifying all qualifying numbers within a set.