To find how many prime numbers exist between the positive integers 60 and 100, we will follow these steps:
### Step 1: Identify the range
We need to find prime numbers between 60 and 100. This means we will consider the numbers from 61 to 99 (since 60 and 100 are not included).
### Step 2: List the numbers in the range
The numbers we will check are:
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.
### Step 3: Check each number for primality
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. We will check each number from our list:
- **61**: Prime
- **62**: Not prime (divisible by 2)
- **63**: Not prime (divisible by 3)
- **64**: Not prime (divisible by 2)
- **65**: Not prime (divisible by 5)
- **66**: Not prime (divisible by 2)
- **67**: Prime
- **68**: Not prime (divisible by 2)
- **69**: Not prime (divisible by 3)
- **70**: Not prime (divisible by 2)
- **71**: Prime
- **72**: Not prime (divisible by 2)
- **73**: Prime
- **74**: Not prime (divisible by 2)
- **75**: Not prime (divisible by 5)
- **76**: Not prime (divisible by 2)
- **77**: Not prime (divisible by 7)
- **78**: Not prime (divisible by 2)
- **79**: Prime
- **80**: Not prime (divisible by 2)
- **81**: Not prime (divisible by 3)
- **82**: Not prime (divisible by 2)
- **83**: Prime
- **84**: Not prime (divisible by 2)
- **85**: Not prime (divisible by 5)
- **86**: Not prime (divisible by 2)
- **87**: Not prime (divisible by 3)
- **88**: Not prime (divisible by 2)
- **89**: Prime
- **90**: Not prime (divisible by 2)
- **91**: Not prime (divisible by 7)
- **92**: Not prime (divisible by 2)
- **93**: Not prime (divisible by 3)
- **94**: Not prime (divisible by 2)
- **95**: Not prime (divisible by 5)
- **96**: Not prime (divisible by 2)
- **97**: Prime
- **98**: Not prime (divisible by 2)
- **99**: Not prime (divisible by 3)
### Step 4: Count the prime numbers
The prime numbers we found between 60 and 100 are:
61, 67, 71, 73, 79, 83, 89, 97.
Counting these, we find there are a total of **8 prime numbers**.
### Final Answer
There are **8 prime numbers** between 60 and 100.
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