Counting Numbers between 1000 and 2500 Neither Divisible by 4 nor 7
In this article, we will explore the process of counting the numbers between 1000 and 2500 that are neither divisible by 4 nor by 7 using the principle of complementary counting. This involves understanding arithmetic sequences and the inclusion-exclusion principle.
Understanding the Range
The range we are considering is from 1000 to 2500, inclusive. To find the total number of integers in this range, we can calculate the difference:
1501 2500 - 1000 1
Hence, there are 1501 integers in this range.
Numbers Divisible by 4
We start by counting the numbers within this range that are divisible by 4. The first multiple of 4 greater than or equal to 1000 is 1000 itself, and the last multiple of 4 less than or equal to 2500 is 2500.
Arithmetic Sequence Formulation
Expressing the nth term of the sequence in general form:
a_n 1000 (n - 1) cdot 4
Solving for n when a_n 2500:
1000 (n - 1) cdot 4 2500 Rightarrow (n - 1) cdot 4 1500 Rightarrow n - 1 375 Rightarrow n 376
Therefore, there are 376 numbers divisible by 4 in the given range.
Numbers Divisible by 7
Next, we count the numbers divisible by 7. The first multiple of 7 greater than or equal to 1000 is 1001, and the last multiple of 7 less than or equal to 2500 is 2499.
Another Arithmetic Sequence
Expressing the nth term of this sequence in general form:
a_n 1001 (n - 1) cdot 7
Solving for n when a_n 2499:
1001 (n - 1) cdot 7 2499 Rightarrow (n - 1) cdot 7 1498 Rightarrow n - 1 214 Rightarrow n 215
Thus, there are 215 numbers divisible by 7 in the given range.
Numbers Divisible by Both 4 and 7 (i.e., 28)
Now, we count the numbers that are divisible by both 4 and 7, which means they are divisible by 28. The first multiple of 28 greater than or equal to 1000 is 1004, and the last multiple of 28 less than or equal to 2500 is 2496.
Final Arithmetic Sequence Analysis
Expressing the nth term of this sequence in general form:
a_n 1004 (n - 1) cdot 28
Solving for n when a_n 2496:
1004 (n - 1) cdot 28 2496 Rightarrow (n - 1) cdot 28 1492 Rightarrow n - 1 53 Rightarrow n 54
This means there are 54 numbers divisible by 28 in the given range.
Applying the Inclusion-Exclusion Principle
Using the inclusion-exclusion principle to find the total numbers divisible by either 4 or 7, we calculate:
N4 cup 7 N4 N7 - N4 cap 7 376 215 - 54 537
Numbers Neither Divisible by 4 nor 7
Finally, we subtract the count of numbers divisible by either 4 or 7 from the total count to find the numbers that are neither divisible by 4 nor by 7:
1501 - 537 964
Thus, there are 964 integers between 1000 and 2500 that are neither divisible by 4 nor by 7.