Finding Prime Numbers Between 100 and 120

This question asks us to find out how many prime numbers exist in the range of numbers starting from 100 up to 120. To answer this, we first need to understand what a prime number is.

What is a Prime Number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Numbers greater than 1 that are not prime are called composite numbers.

Identifying Prime Numbers in the Range [100, 120]

We need to examine each integer from 100 to 120 and determine if it fits the definition of a prime number. We can list the numbers and check for divisibility by small prime numbers (2, 3, 5, 7, 11, etc.). If a number has a divisor other than 1 and itself, it's not prime.

Let's go through the numbers one by one:

• 100: Divisible by 2, 5, 10, etc. (Composite)
• 101: Let's check for divisors other than 1 and 101. We only need to check primes up to $\sqrt{101}$, which is slightly more than 10. Primes to check are 2, 3, 5, 7.
  • Not divisible by 2 (it's odd).
  • Sum of digits $1+0+1=2$, not divisible by 3.
  • Doesn't end in 0 or 5, not divisible by 5.
  • $101 \div 7 = 14$ with a remainder of 3. Not divisible by 7.
  Since it has no small prime divisors up to 10, 101 is likely prime. Let's assume it is prime for this range check. (Prime)
• 102: Divisible by 2 (it's even). (Composite)
• 103: Check primes up to $\sqrt{103}$ (slightly over 10): 2, 3, 5, 7.
  • Not divisible by 2, 3, 5.
  • $103 \div 7 = 14$ with a remainder of 5. Not divisible by 7.
  103 is a Prime number.
• 104: Divisible by 2. (Composite)
• 105: Ends in 5, divisible by 5. (Composite)
• 106: Divisible by 2. (Composite)
• 107: Check primes up to $\sqrt{107}$ (slightly over 10): 2, 3, 5, 7.
  • Not divisible by 2, 3, 5.
  • $107 \div 7 = 15$ with a remainder of 2. Not divisible by 7.
  107 is a Prime number.
• 108: Divisible by 2. (Composite)
• 109: Check primes up to $\sqrt{109}$ (slightly over 10): 2, 3, 5, 7.
  • Not divisible by 2, 3, 5.
  • $109 \div 7 = 15$ with a remainder of 4. Not divisible by 7.
  109 is a Prime number.
• 110: Divisible by 10. (Composite)
• 111: Sum of digits $1+1+1=3$, divisible by 3. (Composite: $111 = 3 \times 37$)
• 112: Divisible by 2. (Composite)
• 113: Check primes up to $\sqrt{113}$ (slightly over 10): 2, 3, 5, 7.
  • Not divisible by 2, 3, 5.
  • $113 \div 7 = 16$ with a remainder of 1. Not divisible by 7.
  113 is a Prime number.
• 114: Divisible by 2. (Composite)
• 115: Ends in 5, divisible by 5. (Composite)
• 116: Divisible by 2. (Composite)
• 117: Sum of digits $1+1+7=9$, divisible by 3. (Composite: $117 = 3 \times 39$)
• 118: Divisible by 2. (Composite)
• 119: Divisible by 7 ($119 = 7 \times 17$). (Composite)
• 120: Divisible by 10. (Composite)

Summary of Numbers and Their Type (100 to 120)

The prime numbers between 100 and 120 are 101, 103, 107, 109, and 113.

Counting the Prime Numbers

Let's count the prime numbers we found:

1. 101
2. 103
3. 107
4. 109
5. 113

There are 5 prime numbers between 100 and 120.

Revision Table: Understanding Prime Numbers

Additional Information: Primality Testing Basics

To check if a number $n$ is prime, you don't need to check for divisibility by every number up to $n-1$. You only need to check for divisibility by prime numbers up to the square root of $n$ ($\sqrt{n}$). If $n$ is not divisible by any prime number less than or equal to $\sqrt{n}$, then $n$ is prime.

For example, to check if 113 is prime, we find $\sqrt{113} \approx 10.6$. We only need to check for divisibility by primes up to 10, which are 2, 3, 5, and 7. As shown above, 113 is not divisible by any of these, so it is prime.