Q3 Write down all of 3 numbers b/w 140-200 which have exactly 3 factors.
Answers
First you should know how to calculate number of factors of any number.
Let’s take 72 for example..
Step 1 : Calculate the prime factorization of number.
According to number theory, every integer (except -1, 0, and 1) has a number of prime numbers that, when multiplied together, will equal the number. Remember that 0 and 1 are not prime numbers.
72 should be factorized into 2 and 36, 2, 6, and 6, and finally, 2, 2, 3, 2, 3, which equals (2^3)*(3^2).
Step 2 :Take all the exponents and add one to each of them.
In the example (2^3) and (3^2), the exponents are 3 and 2 → adding one to each will make them 4 and 3. Let’s call them modified exponents.
Step 3 : Multiply the modified exponents together.
4 x 3 = 12. There are 12 factors for the number 72 --> 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
Now coming to your original question, for a number to have exactly three factors it should be a square of a prime number.
Since you want only three digit number having exactly three factors, therefore
numbers will be 11^2 = 121 , 13^2 = 169 , 17^2 = 289 , 19^2 = 361 , 23^2 = 529 , 29^2 = 841 , 31^2 = 961 .
So, there are seven three digit numbers,exactly having three factors