Question

Find numbers between $ 1 $ and $ 100 $ having exactly three factors.

Answer 1

Hint: Here, in this question we have to find the numbers between $ 1 $ and $ 100 $ which have exactly three factors. Those numbers can be prime numbers, composite numbers of the square of prime numbers. We should know what are prime and composite numbers. After finding the numbers, we will check if they have exactly three factors or not.

Complete step by step solution:
A prime number is a number which is not divisible by any other number except one and itself.
Composite numbers are those natural numbers that are not prime numbers and have more than two positive factors.
Now, according to the question
First, we find out the prime numbers between $ 1 $ and $ 100 $
Some of the prime numbers are $ 2,3,5,7,..... $ . Let us now find their factors.
Factors of $ 2 $ are $ 1,2 $
Factors of $ 3 $ are $ 1,3 $
Similarly, all the prime numbers have exactly two factors, those being $ 1 $ and itself.
Now, we find out the composite numbers between $ 1 $ and $ 100 $
Some of the composite numbers are $ 6,8,12... $ . Let us now find their factors.
Factors of $ 6 $ are $ 1,2,3,6 $
Factors of $ 8 $ are $ 1,2,4,8 $
Similarly, all the composite numbers have more than three factors.
So, we can say that prime numbers have exactly two factors and no composite number has exactly three factors.
Let us check for square of prime numbers which are below $ 100 $ i.e.,
$
   \Rightarrow 4 = 2 \times 2 \\
   \Rightarrow 9 = 3 \times 3 \\
   \Rightarrow 25 = 5 \times 5 \\
   \Rightarrow 49 = 7 \times 7 \\
$
Now we find their factors,
Factors of $ 4 $ are $ 1,2,4 $
Factors of $ 9 $ are $ 1,3,9 $
Factors of $ 25 $ are $ 1,5,25 $
Factors of $ 49 $ are $ 1,7,49 $

Therefore, the numbers between $ 1 $ and $ 100 $ which have exactly three factors are $ 4,9,25,49 $ .

Note: This was an easy question. Students should remember that while expressing the numbers as a product of prime factors, always start with the least number. It will be much easier to find the prime factors. A fun fact- a perfect square is always a composite number. Two composite numbers may be coprime or relatively prime.