How many of these numbers are prime ?? || 99% Cannot Answer This !

Here is a popular Math problem. you really have to invest some time to solve this problem. 99% fail to answer this question because without knowing the trick it is impossible to solve this difficult math problem.

let's get into the details now.

The digits 0-9 (0,1,2,3,4,5,6,7,8,9) can be rearranged into 3,628,800 distinct 10 digits numbers. How many of these numbers are prime?

Let's elaborate on the problem, first of all, it is important to understand what are prime numbers. A prime number is a number that is divisible only by itself and 1 (e.g. 2, 3, 5, 7, 11).  The digits 0-9(0,1,2,3,4,5,6,7,8,9) can be rearranged into many distinct 10 digits numbers Your task is to find all the prime numbers formed using the digits 0-9.

give it a try before checking the solution. 

Solution

Basically, you need a trick to solve this complex problem, otherwise, it is no way possible to solve the problem. It is not easy to find the solution to this problem unless you know the trick to find those numbers that are prime numbers. 

The trick which we can use here is divisibility rules.

The most important rule which comes in handy here is the divisibility rule to find whether the number is divisible by 3 or not. The rule states that a number is divisible by 3 only If the sum of all digits of a number is divisible by 3.

For example, 123 is divisible by 3 because the sum of all the digits in the number is 1 + 2 + 3 = 6 => 6 is divisible by 3 hence 123 is also divisible by 3. 

Take another example 1234 and this is not divisible by 3 because the sum of all the digits in the number 1234 is 10 => which is not divisible by 3. 

This is an interesting rule, regardless of the number of digits in a given number this rule works. We can utilize this rule to solve this problem. I will explain why this rule is important.

What we can do now is, add all the numbers from 0 to 9, it is because regardless of the arrangement of numbers in any combination their sum remains constant. The Sum of all numbers from 0 to 9 is 45 and the sum of the digits in the number 45 results in 9 which is 4+5 = 9, since 9 is divisible by 3 regardless of the arrangement of 10 digits 0-9  in whatever way, the number is divisible by 3.

Thus there are more than 2 factors for any arrangement of digits from 0-9. If the number has more than 2 factors then the number is not a prime number.

Thus answer to this problem is None. None of the numbers are prime when The digits 0-9(0,1,2,3,4,5,6,7,8,9) rearranged in any distinct 10 digits numbers.

to summarize, The sum of numbers from 0-9(0,1,2,3,4,5,6,7,8,9) is 45 and therefore can be divisible by 3. since the number of factors is more than 2, none of the numbers are prime.