If m is the number of prime numbers between 0 and 50; and n is the number of prime numbers between 50 and 100, then what is (m – n) equal to?
5
Finding the Difference Between Counts of Prime Numbers
The question asks us to find the value of \((m - n)\), where \(m\) is the number of prime numbers between 0 and 50, and \(n\) is the number of prime numbers between 50 and 100.
First, let's understand what a prime number is. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Counting Prime Numbers Between 0 and 50 (m)
We need to list all the prime numbers that are greater than 0 and less than 50. Let's list them out:
- 2
- 3
- 5
- 7
- 11
- 13
- 17
- 19
- 23
- 29
- 31
- 37
- 41
- 43
- 47
Counting these numbers, we find there are 15 prime numbers between 0 and 50.
So, \(m = 15\).
Counting Prime Numbers Between 50 and 100 (n)
Next, we need to list all the prime numbers that are greater than 50 and less than 100. Let's list them:
- 53
- 59
- 61
- 67
- 71
- 73
- 79
- 83
- 89
- 97
Counting these numbers, we find there are 10 prime numbers between 50 and 100.
So, \(n = 10\).
Calculating (m - n)
Now we need to find the difference \((m - n)\).
Substitute the values we found for \(m\) and \(n\):
\((m - n) = 15 - 10\)
\((m - n) = 5\)
The difference between the number of prime numbers between 0 and 50 and the number of prime numbers between 50 and 100 is 5.
Comparing with Options
The calculated value is 5, which matches Option 2.
| Range | Prime Numbers | Count |
|---|---|---|
| 0 to 50 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 | \(m = 15\) |
| 50 to 100 | 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 | \(n = 10\) |
Therefore, \(m - n = 15 - 10 = 5\).
Revision Table: Prime Numbers and Counting
| Concept | Description | Example |
|---|---|---|
| Prime Number | A natural number greater than 1 with exactly two distinct positive divisors: 1 and itself. | 2, 3, 5, 7, 11, etc. |
| Composite Number | A natural number greater than 1 that is not prime (i.e., it has more than two positive divisors). | 4 (divisors 1, 2, 4), 6 (divisors 1, 2, 3, 6) |
| Number 1 | Neither prime nor composite. It has only one positive divisor (itself). | - |
Additional Information: Distribution of Prime Numbers
The question highlights that the number of prime numbers is not evenly distributed. There are more prime numbers in the range 0-50 (15 primes) compared to the range 50-100 (10 primes). As numbers get larger, prime numbers generally become less frequent, although there is no simple formula to predict their exact distribution. The study of the distribution of prime numbers is a significant area in number theory.
Some important points about prime numbers:
- 2 is the only even prime number.
- All prime numbers greater than 2 are odd.
- There are infinitely many prime numbers (proven by Euclid).
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