Question

How many numbers are there between 100 and 200 whose only prime factors are 2 and 3?

Answer 1

Hint: The numbers between $100$ and $200$ whose only prime factors are $2$ and $3$ can be calculated by, first write all the numbers between $100$ and $200$, then we will apply the divisibility rule of $2$ and we will apply the divisibility rule of $3$ and then we will find the common numbers between the prime factors of $2$ and the prime factor of $3$ to get the numbers which are the prime factor of both $2$ and $3$ and then we count the common prime factors of both $2$ and $3$ between the numbers $100$ and $200$ to know the numbers between $100$ and $200$ whose only prime factors are $2$ and $3$.

Complete step by step solution:
The numbers between $100$ and $200$ are:
$100, 102, 103, 104, 105, 106, 107, 108, 109, …, 199$
By divisibility rule of $2$:
All even numbers are divisible by $2$
$102, 104, 106, 108, 110, 112,…, 198$
Numbers between the number $100$ and $200$, whose prime factor is $2$ are:
$102, 104, 106, 108, 110, 112, …, 198$
By divisibility rule of $3$:
All numbers whose sum of digits is divisible by $3$ is divisible by $3$
$102, 105, 108, 111, 114, 117, ..., 198$
Numbers between the number $100$ and $200$, whose prime factor is $3$ are:
$102, 105, 108, 111, 114, 117, …, 198$
Now we will find the common numbers between the prime factors $2$ and prime factors of $3$:
$102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198$

$\therefore $ The numbers between 100 and 200 whose only prime factors are 2 and 3 are 17.

Note:
The definition of prime number: The numbers which are only divisible by 1 and the number itself are the prime numbers.
Divisibility rule of 2: Means the numbers whose last digit is even then it is divisible by 2.
Divisibility rule of 3: Means the numbers whose sum of the digit is divisible by 3.