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:
$100,102,103,104,105,106,107,108,109,\dots ,199$
By divisibility rule of $2$:
All even numbers are divisible by $2$
$102,104,106,108,110,112,\dots ,198$
Numbers between the number $100$ and $200$, whose prime factor is $2$ are:
$102,104,106,108,110,112,\dots ,198$
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:
$102,105,108,111,114,117,\dots ,198$
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.