Counting Four-Digit Numbers Not Divisible by 4 or 6: A Comprehensive Guide
In this article, we will provide a detailed step-by-step approach to find how many four-digit numbers are not divisible by 4 or by 6 using the principle of complementary counting. We will follow the principles of arithmetic sequences, divisibility rules, and the inclusion-exclusion principle to solve this problem.
Introduction to the Problem
Understanding the problem of determining how many four-digit numbers are not divisible by 4 or 6 is an excellent exercise in number theory and combinatorics. Complementary counting, an essential technique in combinatorial mathematics, is particularly useful in such scenarios. We will tackle this problem by first identifying how many four-digit numbers are divisible by 4, 6, and both. Then, we will apply the principle of inclusion-exclusion to find the final answer.
Total Number of Four-Digit Numbers
First, let's establish the range of four-digit numbers. The smallest four-digit number is 1000, and the largest is 9999. Thus, the total number of four-digit numbers is:
9999 - 1000 1 9000
Numbers Divisible by 4
A number is divisible by 4 if its last two digits form a number that is divisible by 4. The first four-digit number divisible by 4 is 1000, and the last is 9996. To find how many four-digit numbers are divisible by 4, we consider the arithmetic sequence:
1000, 1004, 1008, ..., 9996
Using the formula for the nth term of an arithmetic sequence, we can find the number of terms (n) in this sequence:
9996 1000 (n - 1) × 4
Solving for n:
n - 1 (9996 - 1000) / 4 8996 / 4 2249
n 2250
Thus, there are 2250 four-digit numbers divisible by 4.
Numbers Divisible by 6
A number is divisible by 6 if it is divisible by both 2 and 3. The first four-digit number divisible by 6 is 1002, and the last is 9996. Using the same method:
9996 1002 (n - 1) × 6
Solving for n:
n - 1 (9996 - 1002) / 6 8994 / 6 1499
n 1500
Thus, there are 1500 four-digit numbers divisible by 6.
Numbers Divisible by Both 4 and 6 (i.e., by 12)
A number is divisible by 12 if it is divisible by both 4 and 3. The first four-digit number divisible by 12 is 1008, and the last is 9996. Using the same method:
9996 1008 (n - 1) × 12
Solving for n:
n - 1 (9996 - 1008) / 12 8988 / 12 749
n 750
Thus, there are 750 four-digit numbers divisible by 12.
Applying the Principle of Inclusion-Exclusion
Now, using the principle of inclusion-exclusion, we can find the total number of four-digit numbers divisible by either 4 or 6:
Divisible by 4 or 6 Divisible by 4 Divisible by 6 - Divisible by 12
2250 1500 - 750 3000
Four-Digit Numbers Not Divisible by 4 or 6
Finally, we subtract the count of four-digit numbers divisible by either 4 or 6 from the total number of four-digit numbers:
Not divisible by 4 or 6 9000 - 3000 6000
Therefore, the number of four-digit numbers that are not divisible by 4 or by 6 is 6000.
Conclusion
By systematically organizing the problem using arithmetic sequences and the inclusion-exclusion principle, we were able to find the number of four-digit numbers not divisible by 4 or 6. This method not only solves the given question but also provides a framework for solving similar problems involving divisibility and combinatorial mathematics.