Math Problem Statement
How many numbers between 1 and 100 (inclusive) are divisible by 4 or 5?
Solution
We need to find how many numbers between and (inclusive) are divisible by or . We use the principle of inclusion-exclusion:
Step 1: Count numbers divisible by 4
A number is divisible by if it is in the sequence: This forms an arithmetic sequence with:
- First term:
- Common difference:
- Last term:
The number of terms is given by:
Step 2: Count numbers divisible by 5
A number is divisible by if it is in the sequence: This forms an arithmetic sequence with:
- First term:
- Common difference:
- Last term:
The number of terms is:
Step 3: Count numbers divisible by both 4 and 5 (i.e., by 20)
A number is divisible by if it is in the sequence: This forms an arithmetic sequence with:
- First term:
- Common difference:
- Last term:
The number of terms is:
Step 4: Apply the Inclusion-Exclusion Principle
Using the formula:
Thus, the number of integers between 1 and 100 (inclusive) that are divisible by 4 or 5 is:
Would you like a more detailed explanation or an alternative method?
Related Questions:
- How many numbers between 1 and 200 are divisible by 6 or 9?
- How many numbers between 1 and 500 are divisible by 7 or 11?
- How many numbers between 1 and 100 are divisible by both 4 and 5?
- What is the sum of all numbers between 1 and 100 that are divisible by 4 or 5?
- How many numbers between 1 and 100 are divisible by neither 4 nor 5?
Tip:
The Inclusion-Exclusion Principle is useful in counting problems where overlapping sets are involved!
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Divisibility
Inclusion-Exclusion Principle
Formulas
Number of terms in an arithmetic sequence: n = (Last term - First term) / Common difference + 1
Inclusion-Exclusion Principle: |A ∪ B| = |A| + |B| - |A ∩ B|
Theorems
Inclusion-Exclusion Principle
Suitable Grade Level
Grades 6-9
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