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 11 and 100100 (inclusive) are divisible by 44 or 55. We use the principle of inclusion-exclusion:

Step 1: Count numbers divisible by 4

A number is divisible by 44 if it is in the sequence: 4,8,12,,1004, 8, 12, \dots, 100 This forms an arithmetic sequence with:

  • First term: 44
  • Common difference: 44
  • Last term: 100100

The number of terms is given by: n=10044+1=964+1=25n = \frac{100 - 4}{4} + 1 = \frac{96}{4} + 1 = 25

Step 2: Count numbers divisible by 5

A number is divisible by 55 if it is in the sequence: 5,10,15,,1005, 10, 15, \dots, 100 This forms an arithmetic sequence with:

  • First term: 55
  • Common difference: 55
  • Last term: 100100

The number of terms is: m=10055+1=955+1=20m = \frac{100 - 5}{5} + 1 = \frac{95}{5} + 1 = 20

Step 3: Count numbers divisible by both 4 and 5 (i.e., by 20)

A number is divisible by 2020 if it is in the sequence: 20,40,60,80,10020, 40, 60, 80, 100 This forms an arithmetic sequence with:

  • First term: 2020
  • Common difference: 2020
  • Last term: 100100

The number of terms is: k=1002020+1=8020+1=5k = \frac{100 - 20}{20} + 1 = \frac{80}{20} + 1 = 5

Step 4: Apply the Inclusion-Exclusion Principle

Using the formula: Total=(Divisible by 4)+(Divisible by 5)(Divisible by 20)\text{Total} = (\text{Divisible by 4}) + (\text{Divisible by 5}) - (\text{Divisible by 20}) =25+205=40= 25 + 20 - 5 = 40

Thus, the number of integers between 1 and 100 (inclusive) that are divisible by 4 or 5 is: 40\boxed{40}

Would you like a more detailed explanation or an alternative method?

Related Questions:

  1. How many numbers between 1 and 200 are divisible by 6 or 9?
  2. How many numbers between 1 and 500 are divisible by 7 or 11?
  3. How many numbers between 1 and 100 are divisible by both 4 and 5?
  4. What is the sum of all numbers between 1 and 100 that are divisible by 4 or 5?
  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